These problems practice solving quadratic equations, following Common Core A-REI.B.4. They start with square roots and the zero-product property, then move to completing the square, the quadratic formula and the discriminant.
The last problems turn real situations into quadratic equations: a poster, a stone, consecutive integers, a garden border and ticket sales. Show every step and check each answer.
Problem 1: Standard form
Write each equation in the form \(ax^{2}+bx+c=0\) and name \(a\), \(b\) and \(c\).
- \(3x^{2}=5x-2\)
- \(x(x+4)=21\)
Problem 2: Perfect-square roots
Solve each equation.
- \(x^{2}=81\)
- \(x^{2}-11=25\)
Problem 3: Square roots, not perfect
Solve each equation. Give exact answers, and round part (a) to the nearest hundredth.
- \(x^{2}=20\)
- \(3x^{2}=75\)
Problem 4: Square both sides back
Solve \((x-3)^{2}=16\). Locate the solutions on the number line.
Problem 5: Square roots with a shift
Solve \((x+2)^{2}=10\). Give exact answers and decimals rounded to the nearest hundredth.
Problem 6: Zero-product property
Solve \((x-4)(x+7)=0\).
Problem 7: Factor a trinomial
Solve \(x^{2}+9x+20=0\) by factoring.
Problem 8: Negative constant
Solve \(x^{2}-3x-28=0\) by factoring.
Problem 9: Common factor
Solve each equation.
- \(x^{2}-6x=0\)
- \(x^{2}+8x=0\)
Problem 10: Rearrange first
Solve \(x^{2}=2x+15\).
Problem 11: Leading coefficient 2
Solve \(2x^{2}+7x+3=0\) by factoring.
Problem 12: Solve from a graph
The graph of \(y=x^{2}-2x-8\) is shown. Use it to solve \(x^{2}-2x-8=0\). Then confirm the answers by factoring.
Problem 13: Complete the square
Solve \(x^{2}+6x+5=0\) by completing the square.
Problem 14: Irrational solutions
Solve \(x^{2}-8x+3=0\) by completing the square. Give exact and rounded answers.
Problem 15: Odd middle coefficient
Solve \(x^{2}+5x-2=0\) by completing the square.
Problem 16: Divide first
Solve \(2x^{2}-12x+10=0\) by completing the square.
Problem 17: Use the formula
Solve \(2x^{2}-5x-3=0\) with the quadratic formula.
Problem 18: Formula with a radical
Solve \(x^{2}-4x-3=0\) with the quadratic formula. Give the exact answer in simplest form and decimals rounded to the nearest hundredth.
Problem 19: Formula, a not 1
Solve \(3x^{2}-4x-2=0\). Round to the nearest hundredth.
Problem 20: Discriminant and graph
The graph of \(y=x^{2}-2x+3\) is shown. How many real solutions does \(x^{2}-2x+3=0\) have? Confirm with the discriminant.
Problem 21: Find k
For what value of \(k\) does \(x^{2}+6x+k=0\) have exactly one real solution? Find that solution.
Problem 22: Rectangle area
The length of a rectangular poster is 3 feet more than its width. Its area is 70 square feet. Find the dimensions.
Problem 23: Stone from a cliff
Priya tosses a stone upward from a ledge. Its height in feet after \(t\) seconds is \(h(t)=-16t^{2}+32t+48\).
- When does the stone hit the ground?
- At what times is the height 48 feet?
Problem 24: Consecutive integers
The product of two consecutive positive integers is 156. Find the integers. Is there another pair of consecutive integers with product 156?
Problem 25: Garden border
Aisha has a garden that is 12 feet by 8 feet. She builds a stone border of uniform width \(x\) around it. The garden and border together have an area of 192 square feet. Find \(x\).
Problem 26: School play tickets
The drama club sells \(200-10x\) tickets when the price is \(x\) dollars. The revenue is \(x(200-10x)\).
- At what price is the revenue \(\$1{,}000\)?
- Can the revenue reach \(\$1{,}200\)? Use the discriminant to decide.
Problem 27: Find the error
Liam solves \(x^{2}-5x=14\) like this: \(x(x-5)=14\), so \(x=14\) or \(x-5=14\). Explain his mistake and solve the equation correctly.
See every chapter on the Algebra 1 math page.