Quadratic Equations - PSAT Math
Card 1 of 30
Find the vertex of $y=(x-3)^2-5$.
Find the vertex of $y=(x-3)^2-5$.
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$(3,-5)$. In vertex form $y=a(x-h)^2+k$, the vertex is $(h,k)$.
$(3,-5)$. In vertex form $y=a(x-h)^2+k$, the vertex is $(h,k)$.
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Find the discriminant of $3x^2-4x+1=0$.
Find the discriminant of $3x^2-4x+1=0$.
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$4$. Calculate $b^2-4ac = (-4)^2-4(3)(1) = 16-12 = 4$.
$4$. Calculate $b^2-4ac = (-4)^2-4(3)(1) = 16-12 = 4$.
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What is the direction of opening for $y=ax^2+bx+c$ when $a>0$?
What is the direction of opening for $y=ax^2+bx+c$ when $a>0$?
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Opens upward. Positive leading coefficient creates a U-shaped parabola.
Opens upward. Positive leading coefficient creates a U-shaped parabola.
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What does $\Delta<0$ tell you about the solutions of $ax^2+bx+c=0$?
What does $\Delta<0$ tell you about the solutions of $ax^2+bx+c=0$?
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No real solutions (two complex solutions). Negative under square root means no real solutions exist.
No real solutions (two complex solutions). Negative under square root means no real solutions exist.
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What does $\Delta=0$ tell you about the solutions of $ax^2+bx+c=0$?
What does $\Delta=0$ tell you about the solutions of $ax^2+bx+c=0$?
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One real solution (a repeated root). Zero discriminant makes $\pm\sqrt{0} = 0$, giving one solution.
One real solution (a repeated root). Zero discriminant makes $\pm\sqrt{0} = 0$, giving one solution.
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What does $\Delta>0$ tell you about the solutions of $ax^2+bx+c=0$?
What does $\Delta>0$ tell you about the solutions of $ax^2+bx+c=0$?
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Two distinct real solutions. Positive discriminant means the square root yields two different values.
Two distinct real solutions. Positive discriminant means the square root yields two different values.
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What is the discriminant of $ax^2+bx+c=0$?
What is the discriminant of $ax^2+bx+c=0$?
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$\Delta=b^2-4ac$. The expression under the square root in the quadratic formula.
$\Delta=b^2-4ac$. The expression under the square root in the quadratic formula.
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What is the quadratic formula for solutions to $ax^2+bx+c=0$?
What is the quadratic formula for solutions to $ax^2+bx+c=0$?
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$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. Derived by completing the square on the general quadratic equation.
$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. Derived by completing the square on the general quadratic equation.
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What is the standard form of a quadratic equation in one variable?
What is the standard form of a quadratic equation in one variable?
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$ax^2+bx+c=0$, where $a\ne 0$. Must have $a \ne 0$ to ensure the equation is quadratic, not linear.
$ax^2+bx+c=0$, where $a\ne 0$. Must have $a \ne 0$ to ensure the equation is quadratic, not linear.
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Identify the solutions of $x^2-5x+6=0$.
Identify the solutions of $x^2-5x+6=0$.
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$x=2$ and $x=3$. Factor as $(x-2)(x-3)=0$, so $x=2$ or $x=3$.
$x=2$ and $x=3$. Factor as $(x-2)(x-3)=0$, so $x=2$ or $x=3$.
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What is the factored form of a quadratic with zeros $r_1$ and $r_2$?
What is the factored form of a quadratic with zeros $r_1$ and $r_2$?
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$y=a(x-r_1)(x-r_2)$. Each factor equals zero when $x$ equals a root.
$y=a(x-r_1)(x-r_2)$. Each factor equals zero when $x$ equals a root.
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What is the $y$-intercept of $y=ax^2+bx+c$?
What is the $y$-intercept of $y=ax^2+bx+c$?
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$(0,c)$. Found by substituting $x=0$ into the equation.
$(0,c)$. Found by substituting $x=0$ into the equation.
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What is the vertex of $y=a(x-h)^2+k$?
What is the vertex of $y=a(x-h)^2+k$?
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$(h,k)$. The vertex is the point where the parabola reaches its extreme value.
$(h,k)$. The vertex is the point where the parabola reaches its extreme value.
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What is the $x$-coordinate of the vertex for $y=ax^2+bx+c$?
What is the $x$-coordinate of the vertex for $y=ax^2+bx+c$?
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$x=-\frac{b}{2a}$. Found by completing the square or using calculus to find the minimum/maximum.
$x=-\frac{b}{2a}$. Found by completing the square or using calculus to find the minimum/maximum.
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What is the vertex form of a quadratic function?
What is the vertex form of a quadratic function?
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$y=a(x-h)^2+k$. Shows the vertex at $(h,k)$ and vertical stretch/compression by $a$.
$y=a(x-h)^2+k$. Shows the vertex at $(h,k)$ and vertical stretch/compression by $a$.
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What is the discriminant of $3x^2-6x+1=0$?
What is the discriminant of $3x^2-6x+1=0$?
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$24$. $\Delta = (-6)^2 - 4(3)(1) = 36 - 12 = 24$.
$24$. $\Delta = (-6)^2 - 4(3)(1) = 36 - 12 = 24$.
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Identify the solutions of $2x^2+3x-2=0$.
Identify the solutions of $2x^2+3x-2=0$.
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$x=\frac{1}{2}$ and $x=-2$. Factor as $(2x-1)(x+2)=0$, so $x=\frac{1}{2}$ or $x=-2$.
$x=\frac{1}{2}$ and $x=-2$. Factor as $(2x-1)(x+2)=0$, so $x=\frac{1}{2}$ or $x=-2$.
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What is the axis of symmetry for $y=2x^2-8x+5$?
What is the axis of symmetry for $y=2x^2-8x+5$?
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$x=2$. Use $x = -\frac{b}{2a} = -\frac{-8}{2(2)} = \frac{8}{4} = 2$.
$x=2$. Use $x = -\frac{b}{2a} = -\frac{-8}{2(2)} = \frac{8}{4} = 2$.
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What is the direction of opening for $y=ax^2+bx+c$ when $a<0$?
What is the direction of opening for $y=ax^2+bx+c$ when $a<0$?
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Opens downward. Negative leading coefficient creates an inverted U-shape.
Opens downward. Negative leading coefficient creates an inverted U-shape.
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Find the axis of symmetry of $y=2x^2-8x+1$.
Find the axis of symmetry of $y=2x^2-8x+1$.
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$x=2$. Use $x=-\frac{b}{2a} = -\frac{-8}{2(2)} = \frac{8}{4} = 2$.
$x=2$. Use $x=-\frac{b}{2a} = -\frac{-8}{2(2)} = \frac{8}{4} = 2$.
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Identify the direction the parabola opens for $y=ax^2+bx+c$ when $a<0$.
Identify the direction the parabola opens for $y=ax^2+bx+c$ when $a<0$.
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Opens downward. Negative leading coefficient creates an inverted U-shape.
Opens downward. Negative leading coefficient creates an inverted U-shape.
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What is the product of the solutions of $ax^2+bx+c=0$ in terms of $a$ and $c$?
What is the product of the solutions of $ax^2+bx+c=0$ in terms of $a$ and $c$?
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$r_1r_2=\frac{c}{a}$. Another application of Vieta's formulas.
$r_1r_2=\frac{c}{a}$. Another application of Vieta's formulas.
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What condition on $b^2-4ac$ gives no real solutions?
What condition on $b^2-4ac$ gives no real solutions?
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$b^2-4ac<0$. Negative discriminant means no real square root exists.
$b^2-4ac<0$. Negative discriminant means no real square root exists.
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What is the sum of the solutions of $ax^2+bx+c=0$ in terms of $a$ and $b$?
What is the sum of the solutions of $ax^2+bx+c=0$ in terms of $a$ and $b$?
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$r_1+r_2=-\frac{b}{a}$. From Vieta's formulas for quadratic equations.
$r_1+r_2=-\frac{b}{a}$. From Vieta's formulas for quadratic equations.
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What is the vertex form of a quadratic function?
What is the vertex form of a quadratic function?
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$y=a(x-h)^2+k$. Shows the vertex at $(h,k)$ and vertical stretch/compression by $a$.
$y=a(x-h)^2+k$. Shows the vertex at $(h,k)$ and vertical stretch/compression by $a$.
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What condition on $b^2-4ac$ gives exactly one real solution (a double root)?
What condition on $b^2-4ac$ gives exactly one real solution (a double root)?
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$b^2-4ac=0$. Zero discriminant makes $\pm\sqrt{0} = 0$, giving one repeated solution.
$b^2-4ac=0$. Zero discriminant makes $\pm\sqrt{0} = 0$, giving one repeated solution.
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What condition on $b^2-4ac$ gives exactly two distinct real solutions?
What condition on $b^2-4ac$ gives exactly two distinct real solutions?
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$b^2-4ac>0$. Positive discriminant means the square root yields two different values.
$b^2-4ac>0$. Positive discriminant means the square root yields two different values.
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What is the discriminant for $ax^2+bx+c=0$?
What is the discriminant for $ax^2+bx+c=0$?
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$b^2-4ac$. The expression under the square root in the quadratic formula.
$b^2-4ac$. The expression under the square root in the quadratic formula.
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State the quadratic formula for solutions of $ax^2+bx+c=0$.
State the quadratic formula for solutions of $ax^2+bx+c=0$.
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$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. Derived by completing the square on the general quadratic equation.
$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. Derived by completing the square on the general quadratic equation.
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What is the standard form of a quadratic equation in one variable?
What is the standard form of a quadratic equation in one variable?
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$ax^2+bx+c=0$ with $a\ne 0$. Must have $a \ne 0$ to ensure the equation is quadratic, not linear.
$ax^2+bx+c=0$ with $a\ne 0$. Must have $a \ne 0$ to ensure the equation is quadratic, not linear.
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