Solving Quadratic Equations: math practice, Grade 9 – download the PDF

1 Square root method, first steps ★★★

Solve each equation.

1. \(x^2=49\)
2. \(5x^2=45\)

2 A square plus a shift ★★★

Solve \((x+4)^2=25\).

3 Answer as a radical ★★★

Solve \(x^2-20=0\). Give the exact answers, then round to the nearest hundredth.

4 True or false? ★★★

Maya says: “The equation \(x^2=-9\) has the two solutions \(3\) and \(-3\).” Is she right? Explain.

5 Standard form ★★★

Write each equation in standard form \(ax^2+bx+c=0\) and give \(a\), \(b\), \(c\).

1. \(3x^2=2x+7\)
2. \((x+3)(x-1)=5\)

6 Counting solutions ★★★

Compute the discriminant and say how many real solutions each equation has.

1. \(x^2+5x+2=0\)
2. \(4x^2-12x+9=0\)
3. \(2x^2+x+3=0\)

7 Zero-product warm-up ★★★

Solve \((x-6)(x+2)=0\), then expand it and check that you get \(x^2-4x-12=0\).

8 Complete the square ★★★

Solve \(x^2+8x+7=0\) by completing the square.

9 Completing the square with a radical ★★★

Solve \(x^2-10x+18=0\) by completing the square. Round to the nearest hundredth.

10 Quadratic formula with a leading coefficient ★★★

Solve \(3x^2+x-10=0\) with the quadratic formula.

11 Irrational solutions ★★★

Solve \(x^2-6x+7=0\). Give exact answers and approximations to the nearest hundredth.

12 Which method? ★★★

For each equation, name the method you would pick and solve it.

1. \(2x^2-18=0\)
2. \(x^2+7x+12=0\)
3. \(x^2-3x-5=0\)

13 A rectangular patio ★★★

A rectangular patio is 5 feet longer than it is wide. Its area is 84 square feet. Find its dimensions.

14 Find the error ★★★

A student solves \(x^2+3x=10\) like this: “\(x(x+3)=10\), so \(x=10\) or \(x+3=10\), hence \(x=10\) or \(x=7\).” Explain the mistake and give the correct solutions.

15 Reading a table of values ★★★

Here are values of \(y=x^2-4x+3\).

1. Give the solutions of \(x^2-4x+3=0\).
2. What is the lowest point of the parabola?

16 One solution exactly ★★★

For which values of \(k\) does \(x^2+kx+16=0\) have exactly one real solution? Find that solution when \(k\) is positive.

17 Completing the square when a is not 1 ★★★

Solve \(2x^2-12x+5=0\) by completing the square, then check with the quadratic formula.

18 The falling tool ★★★

A tool slips off scaffolding 144 feet above the ground. Its height after \(t\) seconds is \(h=144-16t^2\). When does it hit the ground, and how high is it after 1 second?

19 A ball thrown upward ★★★

A ball is thrown upward from a ledge. Its height in feet after \(t\) seconds is \(h=-16t^2+32t+48\).

1. When does it hit the ground?
2. When is it back at the height of the ledge?

20 A line cuts a parabola ★★★

Solve the system \(y=x^2-3x\) and \(y=2x-6\).

21 Tangent line or no intersection ★★★

Decide how many points the parabola and the line share, and find them.

1. \(y=x^2+2x+1\) and \(y=4x\)
2. \(y=x^2+1\) and \(y=2x-3\)

22 Reading a graph ★★★

The graph shows \(y=x^2+2x-8\).

1. Solve \(x^2+2x-8=0\).
2. Solve \(x^2+2x-8=-8\) algebraically and explain what it means on the graph.