Quadratic Equations - SAT Math
Card 1 of 32
What is the quadratic formula?
What is the quadratic formula?
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$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. Derived from completing the square on the general quadratic equation.
$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. Derived from completing the square on the general quadratic equation.
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State the quadratic formula.
State the quadratic formula.
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$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. Solves any quadratic by substituting coefficients $a$, $b$, and $c$.
$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. Solves any quadratic by substituting coefficients $a$, $b$, and $c$.
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What is the standard form of a quadratic equation?
What is the standard form of a quadratic equation?
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$ax^2 + bx + c = 0$. General form where $a \neq 0$ determines parabola shape.
$ax^2 + bx + c = 0$. General form where $a \neq 0$ determines parabola shape.
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What is the axis of symmetry for $y = ax^2 + bx + c$?
What is the axis of symmetry for $y = ax^2 + bx + c$?
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$x = \frac{-b}{2a}$. The $x$-coordinate of the vertex, found by calculus or completing the square.
$x = \frac{-b}{2a}$. The $x$-coordinate of the vertex, found by calculus or completing the square.
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What is the vertex of $y = 2(x-1)^2 + 3$?
What is the vertex of $y = 2(x-1)^2 + 3$?
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Vertex: $(1, 3)$. In vertex form $y = a(x-h)^2 + k$, vertex is $(h,k)$.
Vertex: $(1, 3)$. In vertex form $y = a(x-h)^2 + k$, vertex is $(h,k)$.
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For $y = x^2 + 4x + 4$, what is the vertex?
For $y = x^2 + 4x + 4$, what is the vertex?
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Vertex: $(-2, 0)$. This is $(x+2)^2$, so vertex is at $x = -2$, $y = 0$.
Vertex: $(-2, 0)$. This is $(x+2)^2$, so vertex is at $x = -2$, $y = 0$.
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Which method can solve any quadratic equation?
Which method can solve any quadratic equation?
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The quadratic formula. Works for any quadratic, unlike factoring or square roots.
The quadratic formula. Works for any quadratic, unlike factoring or square roots.
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Find the axis of symmetry for $y = 3x^2 + 6x + 1$.
Find the axis of symmetry for $y = 3x^2 + 6x + 1$.
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$x = -1$. Use $x = -\frac{b}{2a} = -\frac{6}{2(3)} = -1$.
$x = -1$. Use $x = -\frac{b}{2a} = -\frac{6}{2(3)} = -1$.
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Determine the parabola direction for $y = 5x^2 - 3x + 2$.
Determine the parabola direction for $y = 5x^2 - 3x + 2$.
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Opens upwards. Positive coefficient of $x^2$ means parabola opens upward.
Opens upwards. Positive coefficient of $x^2$ means parabola opens upward.
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Find the x-intercepts of $y = x^2 - 4$.
Find the x-intercepts of $y = x^2 - 4$.
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$x = -2, x = 2$. Set $y = 0$: $x^2 - 4 = 0$, so $x^2 = 4$.
$x = -2, x = 2$. Set $y = 0$: $x^2 - 4 = 0$, so $x^2 = 4$.
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What is the effect of increasing $a$ in $y = ax^2$ on the parabola?
What is the effect of increasing $a$ in $y = ax^2$ on the parabola?
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Narrower parabola. Larger $|a|$ values compress the parabola horizontally.
Narrower parabola. Larger $|a|$ values compress the parabola horizontally.
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Write $x^2 + 8x + 16$ as a square.
Write $x^2 + 8x + 16$ as a square.
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$(x + 4)^2$. Perfect square trinomial with $a = x$, $b = 4$.
$(x + 4)^2$. Perfect square trinomial with $a = x$, $b = 4$.
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Solve $3x^2 - 12 = 0$.
Solve $3x^2 - 12 = 0$.
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$x = 2, x = -2$. Divide by 3: $x^2 - 4 = 0$, so $x = \pm 2$.
$x = 2, x = -2$. Divide by 3: $x^2 - 4 = 0$, so $x = \pm 2$.
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Identify the $y$-intercept of $y = 2x^2 + 3x + 1$.
Identify the $y$-intercept of $y = 2x^2 + 3x + 1$.
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Intercept: $(0, 1)$. Set $x = 0$ to find where the parabola crosses the $y$-axis.
Intercept: $(0, 1)$. Set $x = 0$ to find where the parabola crosses the $y$-axis.
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Convert $y = x^2 - 4x + 4$ to vertex form.
Convert $y = x^2 - 4x + 4$ to vertex form.
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$y = (x-2)^2$. This is $(x-2)^2$ expanded, so vertex form shows vertex $(2,0)$.
$y = (x-2)^2$. This is $(x-2)^2$ expanded, so vertex form shows vertex $(2,0)$.
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Find the discriminant for $2x^2 - 4x + 2 = 0$.
Find the discriminant for $2x^2 - 4x + 2 = 0$.
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Discriminant: $0$. Calculate $b^2 - 4ac = 16 - 16 = 0$.
Discriminant: $0$. Calculate $b^2 - 4ac = 16 - 16 = 0$.
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What is the standard form of a quadratic equation?
What is the standard form of a quadratic equation?
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$ax^2 + bx + c = 0$. The general form where $a \neq 0$ defines a parabola.
$ax^2 + bx + c = 0$. The general form where $a \neq 0$ defines a parabola.
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Solve for $x$: $x^2 + 2x - 8 = 0$.
Solve for $x$: $x^2 + 2x - 8 = 0$.
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$x = 2, x = -4$. Factor as $(x-2)(x+4) = 0$ or use the quadratic formula.
$x = 2, x = -4$. Factor as $(x-2)(x+4) = 0$ or use the quadratic formula.
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State the condition for a perfect square trinomial.
State the condition for a perfect square trinomial.
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$b^2 = 4ac$. When discriminant equals zero, the quadratic is a perfect square.
$b^2 = 4ac$. When discriminant equals zero, the quadratic is a perfect square.
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Convert $y = x^2 + 6x + 8$ to vertex form.
Convert $y = x^2 + 6x + 8$ to vertex form.
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$y = (x+3)^2 - 1$. Complete the square: $(x+3)^2 = x^2 + 6x + 9$, so subtract 1.
$y = (x+3)^2 - 1$. Complete the square: $(x+3)^2 = x^2 + 6x + 9$, so subtract 1.
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Simplify: $(x + 2)^2$.
Simplify: $(x + 2)^2$.
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$x^2 + 4x + 4$. Apply the perfect square formula $(a+b)^2 = a^2 + 2ab + b^2$.
$x^2 + 4x + 4$. Apply the perfect square formula $(a+b)^2 = a^2 + 2ab + b^2$.
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State the zero-product property.
State the zero-product property.
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If $ab = 0$, then $a = 0$ or $b = 0$. If a product equals zero, at least one factor must be zero.
If $ab = 0$, then $a = 0$ or $b = 0$. If a product equals zero, at least one factor must be zero.
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What is the sum of the roots of $ax^2 + bx + c = 0$?
What is the sum of the roots of $ax^2 + bx + c = 0$?
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$-\frac{b}{a}$. By Vieta's formulas relating coefficients to roots.
$-\frac{b}{a}$. By Vieta's formulas relating coefficients to roots.
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What type of parabola does $y = -x^2$ represent?
What type of parabola does $y = -x^2$ represent?
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A downward-opening parabola. Negative coefficient of $x^2$ makes the parabola open downward.
A downward-opening parabola. Negative coefficient of $x^2$ makes the parabola open downward.
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Factor $x^2 - 9$.
Factor $x^2 - 9$.
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$(x - 3)(x + 3)$. Difference of squares: $a^2 - b^2 = (a-b)(a+b)$.
$(x - 3)(x + 3)$. Difference of squares: $a^2 - b^2 = (a-b)(a+b)$.
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Find the vertex of $y = -x^2 + 4x - 3$.
Find the vertex of $y = -x^2 + 4x - 3$.
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Vertex: $(2, 1)$. Use $x = -\frac{b}{2a} = 2$, then $y = -(4) + 8 - 3 = 1$.
Vertex: $(2, 1)$. Use $x = -\frac{b}{2a} = 2$, then $y = -(4) + 8 - 3 = 1$.
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Find the roots of $x^2 - 5x + 6 = 0$.
Find the roots of $x^2 - 5x + 6 = 0$.
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$x = 2, x = 3$. Factor as $(x-2)(x-3) = 0$ or use the quadratic formula.
$x = 2, x = 3$. Factor as $(x-2)(x-3) = 0$ or use the quadratic formula.
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What is the product of the roots of $ax^2 + bx + c = 0$?
What is the product of the roots of $ax^2 + bx + c = 0$?
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$\frac{c}{a}$. By Vieta's formulas relating coefficients to roots.
$\frac{c}{a}$. By Vieta's formulas relating coefficients to roots.
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What is the minimum value of $y = x^2 - 6x + 9$?
What is the minimum value of $y = x^2 - 6x + 9$?
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Minimum: $0$. This is $(x-3)^2$, so minimum occurs at vertex $(3,0)$.
Minimum: $0$. This is $(x-3)^2$, so minimum occurs at vertex $(3,0)$.
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Identify the vertex form of a quadratic equation.
Identify the vertex form of a quadratic equation.
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$y = a(x-h)^2 + k$. Shows vertex $(h,k)$ and vertical stretch/compression factor $a$.
$y = a(x-h)^2 + k$. Shows vertex $(h,k)$ and vertical stretch/compression factor $a$.
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How do you determine the number of real roots using the discriminant?
How do you determine the number of real roots using the discriminant?
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If $b^2 - 4ac > 0$, 2 real roots; $=0$, 1 real root; $<0$, no real roots. The discriminant determines whether roots are real or complex.
If $b^2 - 4ac > 0$, 2 real roots; $=0$, 1 real root; $<0$, no real roots. The discriminant determines whether roots are real or complex.
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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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$b^2 - 4ac$. 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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