Quadratics & Polynomials - ACT Math
Card 1 of 30
What is the sum of the solutions to $2x^2 - 8x = 0$?
What is the sum of the solutions to $2x^2 - 8x = 0$?
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Sum is $4$. Factor as $2x(x-4) = 0$, roots are $x=0,4$; sum is $0+4=4$.
Sum is $4$. Factor as $2x(x-4) = 0$, roots are $x=0,4$; sum is $0+4=4$.
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What is the expanded form of $(x + 3)^2$?
What is the expanded form of $(x + 3)^2$?
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$x^2 + 6x + 9$. Perfect square: $(a + b)^2 = a^2 + 2ab + b^2$.
$x^2 + 6x + 9$. Perfect square: $(a + b)^2 = a^2 + 2ab + b^2$.
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State the difference of squares factorization formula.
State the difference of squares factorization formula.
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$a^2-b^2=(a-b)(a+b)$. Special factoring pattern for expressions with squared terms.
$a^2-b^2=(a-b)(a+b)$. Special factoring pattern for expressions with squared terms.
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What is the sum of the coefficients in $2x^2 - 3x + 5$?
What is the sum of the coefficients in $2x^2 - 3x + 5$?
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- Add all coefficients: $2 + (-3) + 5 = 4$.
- Add all coefficients: $2 + (-3) + 5 = 4$.
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Identify the leading coefficient in $4x^3 - x^2 + 5$.
Identify the leading coefficient in $4x^3 - x^2 + 5$.
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- The coefficient of the term with the highest degree.
- The coefficient of the term with the highest degree.
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What does $\Delta>0$ indicate about the real solutions of $ax^2+bx+c=0$?
What does $\Delta>0$ indicate about the real solutions of $ax^2+bx+c=0$?
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Two distinct real solutions. Positive discriminant means the parabola crosses the x-axis twice.
Two distinct real solutions. Positive discriminant means the parabola crosses the x-axis twice.
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State the remainder theorem for dividing $f(x)$ by $(x-a)$.
State the remainder theorem for dividing $f(x)$ by $(x-a)$.
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Remainder $=f(a)$. The remainder equals the function value at the divisor's root.
Remainder $=f(a)$. The remainder equals the function value at the divisor's root.
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What does $\Delta<0$ indicate about the solutions of $ax^2+bx+c=0$?
What does $\Delta<0$ indicate about the solutions of $ax^2+bx+c=0$?
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No real solutions (two complex solutions). Negative discriminant means the parabola doesn't cross the x-axis.
No real solutions (two complex solutions). Negative discriminant means the parabola doesn't cross the x-axis.
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In $y=a(x-h)^2+k$, what are the vertex coordinates?
In $y=a(x-h)^2+k$, what are the vertex coordinates?
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Vertex $=(h,k)$. The vertex is at the point $(h,k)$ in this form.
Vertex $=(h,k)$. The vertex is at the point $(h,k)$ in this form.
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Factor completely: $x^2+10x+25$.
Factor completely: $x^2+10x+25$.
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$(x+5)^2$. Perfect square trinomial: $a^2+2ab+b^2=(a+b)^2$.
$(x+5)^2$. Perfect square trinomial: $a^2+2ab+b^2=(a+b)^2$.
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State the definition of the degree of a polynomial.
State the definition of the degree of a polynomial.
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Largest exponent of $x$ with nonzero coefficient. The highest power of the variable with a non-zero coefficient.
Largest exponent of $x$ with nonzero coefficient. The highest power of the variable with a non-zero coefficient.
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State Vieta’s formulas for $ax^2+bx+c=0$ with roots $r_1,r_2$.
State Vieta’s formulas for $ax^2+bx+c=0$ with roots $r_1,r_2$.
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$r_1+r_2=-\frac{b}{a}$ and $r_1r_2=\frac{c}{a}$. Relationships between coefficients and roots without solving the equation.
$r_1+r_2=-\frac{b}{a}$ and $r_1r_2=\frac{c}{a}$. Relationships between coefficients and roots without solving the equation.
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Simplify $2x^2 - 4x + 2$. What is the expression?
Simplify $2x^2 - 4x + 2$. What is the expression?
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$2(x^2 - 2x + 1)$. Factor out common factor of 2.
$2(x^2 - 2x + 1)$. Factor out common factor of 2.
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Solve for $x$: $x^2-5x+6=0$.
Solve for $x$: $x^2-5x+6=0$.
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$x=2$ or $x=3$. Factor as $(x-2)(x-3)=0$ using zero-product property.
$x=2$ or $x=3$. Factor as $(x-2)(x-3)=0$ using zero-product property.
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Solve for $x$: $2x^2+7x+3=0$.
Solve for $x$: $2x^2+7x+3=0$.
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$x=-3$ or $x=-\frac{1}{2}$. Factor as $(2x+1)(x+3)=0$ and solve each factor.
$x=-3$ or $x=-\frac{1}{2}$. Factor as $(2x+1)(x+3)=0$ and solve each factor.
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Factor completely: $x^2-9$.
Factor completely: $x^2-9$.
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$(x-3)(x+3)$. Difference of squares pattern: $a^2-b^2=(a-b)(a+b)$.
$(x-3)(x+3)$. Difference of squares pattern: $a^2-b^2=(a-b)(a+b)$.
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Factor completely: $x^2+10x+25$.
Factor completely: $x^2+10x+25$.
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$(x+5)^2$. Perfect square trinomial: $a^2+2ab+b^2=(a+b)^2$.
$(x+5)^2$. Perfect square trinomial: $a^2+2ab+b^2=(a+b)^2$.
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Expand and simplify: $(x-4)^2$.
Expand and simplify: $(x-4)^2$.
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$x^2-8x+16$. Use the formula $(a-b)^2=a^2-2ab+b^2$.
$x^2-8x+16$. Use the formula $(a-b)^2=a^2-2ab+b^2$.
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Simplify: $(x+2)(x-5)$.
Simplify: $(x+2)(x-5)$.
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$x^2-3x-10$. Use FOIL: $(x+2)(x-5)=x^2-5x+2x-10$.
$x^2-3x-10$. Use FOIL: $(x+2)(x-5)=x^2-5x+2x-10$.
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Find the discriminant of $3x^2-4x+5=0$.
Find the discriminant of $3x^2-4x+5=0$.
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$\Delta=-44$. Calculate $\Delta=(-4)^2-4(3)(5)=16-60=-44$.
$\Delta=-44$. Calculate $\Delta=(-4)^2-4(3)(5)=16-60=-44$.
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Which term is the quadratic term in $5x^2 - 3x + 8$?
Which term is the quadratic term in $5x^2 - 3x + 8$?
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Quadratic term is $5x^2$. Term with $x^2$ (degree 2).
Quadratic term is $5x^2$. Term with $x^2$ (degree 2).
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Evaluate $x^2 - 4x + 4$ for $x = 2$.
Evaluate $x^2 - 4x + 4$ for $x = 2$.
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Result is $0$. Substitute $x=2$: $(2)^2 - 4(2) + 4 = 4 - 8 + 4 = 0$.
Result is $0$. Substitute $x=2$: $(2)^2 - 4(2) + 4 = 4 - 8 + 4 = 0$.
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Find the remainder when $x^3 - 2x^2 + x - 1$ is divided by $x - 1$.
Find the remainder when $x^3 - 2x^2 + x - 1$ is divided by $x - 1$.
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Remainder is $-1$. By remainder theorem: $f(1) = 1 - 2 + 1 - 1 = -1$.
Remainder is $-1$. By remainder theorem: $f(1) = 1 - 2 + 1 - 1 = -1$.
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Determine if $x = -2$ is a root of $x^3 + 2x^2 - 5$.
Determine if $x = -2$ is a root of $x^3 + 2x^2 - 5$.
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Not a root. Substitute $x=-2$: $(-2)^3 + 2(-2)^2 - 5 = -8 + 8 - 5 = -5 \neq 0$.
Not a root. Substitute $x=-2$: $(-2)^3 + 2(-2)^2 - 5 = -8 + 8 - 5 = -5 \neq 0$.
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What is the y-intercept of $y = 3x^2 + 2x - 4$?
What is the y-intercept of $y = 3x^2 + 2x - 4$?
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Y-intercept is $-4$. Y-intercept occurs when $x=0$, giving $y=c$.
Y-intercept is $-4$. Y-intercept occurs when $x=0$, giving $y=c$.
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If $p(x) = x^3 - 2x^2 + 4$, what is $p(1)$?
If $p(x) = x^3 - 2x^2 + 4$, what is $p(1)$?
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$p(1) = 3$. Substitute $x=1$: $(1)^3 - 2(1)^2 + 4 = 1 - 2 + 4 = 3$.
$p(1) = 3$. Substitute $x=1$: $(1)^3 - 2(1)^2 + 4 = 1 - 2 + 4 = 3$.
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What is the vertex of $y = (x-2)^2 + 3$?
What is the vertex of $y = (x-2)^2 + 3$?
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Vertex is $(2, 3)$. In vertex form, $(h,k)$ gives the vertex coordinates.
Vertex is $(2, 3)$. In vertex form, $(h,k)$ gives the vertex coordinates.
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Simplify $(2x - 3)^2$. What is the expression?
Simplify $(2x - 3)^2$. What is the expression?
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$4x^2 - 12x + 9$. Perfect square trinomial: $(a-b)^2 = a^2 - 2ab + b^2$.
$4x^2 - 12x + 9$. Perfect square trinomial: $(a-b)^2 = a^2 - 2ab + b^2$.
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Simplify $(x + 1)(x - 1)$. What is the expression?
Simplify $(x + 1)(x - 1)$. What is the expression?
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$x^2 - 1$. Difference of squares formula.
$x^2 - 1$. Difference of squares formula.
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Factor $x^2 + 5x + 6$ completely.
Factor $x^2 + 5x + 6$ completely.
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$(x + 2)(x + 3)$. Find factors of 6 that add to 5: 2 and 3.
$(x + 2)(x + 3)$. Find factors of 6 that add to 5: 2 and 3.
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