Solving Quadratic Equations: Algebra 1 Math Test Answer Key – PDF Download

Question 1: Square roots (2 points)

1. \(x^{2}=16\), so \(x=\pm 4\).
2. \(x+1=\pm 5\), so \(x=4\) or \(x=-6\). \(x=4\) or \(x=-6\).

Scoring: 1 point for each part; in (a) both signs are needed, in (b) both solutions are needed.

Question 2: Factoring (2 points)

1. \((x+8)(x-5)=0\). \(x=-8\) or \(x=5\).
2. \((3x+2)(x-1)=0\). \(x=-\frac{2}{3}\) or \(x=1\).

Scoring: 1 point for each part, awarded for both solutions.

Question 3: Complete the square (3 points)

\(x^{2}+6x=2\). Add \(3^{2}=9\): \((x+3)^{2}=11\). Then \(x+3=\pm\sqrt{11}\).

Exact: \(x=-3\pm\sqrt{11}\). Rounded: \(x\approx 0.32\) or \(x\approx -6.32\).

Scoring: 1 point for adding 9 to both sides; 1 point for \((x+3)^{2}=11\) and the exact solutions; 1 point for the rounded values.

Question 4: Quadratic formula (3 points)

\(a=2\), \(b=-3\), \(c=-4\). \(b^{2}-4ac=9+32=41\).

\(x=\frac{3\pm\sqrt{41}}{4}\). Rounded: \(x\approx 2.35\) or \(x\approx -0.85\).

Scoring: 1 point for the correct discriminant; 1 point for the exact solutions; 1 point for the rounded values.

Question 5: Discriminant (2 points)

1. \((-6)^{2}-4(3)(4)=36-48=-12\). It is negative, so there are no real solutions.
2. \(64-4k=0\), so \(k=16\).

Scoring: 1 point for each part; in (a) the value and the conclusion are both needed.

Question 6: Use a graph (3 points)

1. The x-intercepts give \(x=-1\) and \(x=5\). Check: \((x-5)(x+1)=x^{2}-4x-5\).
2. Vertex \((2, -9)\); axis \(x=2\).
3. The graph is below the axis between the zeros: \(-1 < x < 5\).

Scoring: 1 point for each part.

Question 7: Rectangle (2 points)

\(w(w+5)=84\), so \(w^{2}+5w-84=0\) and \((w+12)(w-7)=0\). The solution \(w=-12\) is rejected, so the width is 7 ft and the length is 12 ft. Check: \(7\times 12=84\).

Scoring: 1 point for the equation and its solutions; 1 point for rejecting \(-12\) and giving both dimensions.

Question 8: Ball flight (2 points)

1. \(-16t^{2}+40t+24=0\) gives \(-8(2t+1)(t-3)=0\), so \(t=-\frac{1}{2}\) or \(t=3\). The ball lands after 3 seconds.
2. \(t=-\frac{1}{2}\) is before the ball was hit, and time cannot be negative.

Scoring: 1 point for solving correctly and giving 3 seconds; 1 point for a valid reason to reject \(t=-\frac{1}{2}\).

Question 9: No real solution (1 point)

Squaring any real number gives a result that is zero or positive. \(3^{2}=9\) and \((-3)^{2}=9\), not \(-9\). So \(x^{2}=-9\) has no real solutions.

Scoring: 1 point for stating that no real number squares to a negative value (or checking that \(3^{2}=9\)).

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