Chapter 14

Taking on Quadratic Equations

A quadratic equation has the form ax2 + bx + c = 0. The equation can have exactly two solutions, only one solution (a double root), or no solutions among the real numbers. Where no real solution occurs, imaginary numbers are brought into the picture. Quadratic equations are solved most easily when the expression that's set to 0 factors, but the quadratic formula is also a nice means to finding solutions.

The Problems You'll Work On

In this chapter, you work with quadratic equations in the following ways:

  • Applying the square root rule
  • Solving equations by using factoring and the multiplication property of 0
  • Using the quadratic formula and simplifying radicals when possible
  • Solving quadratic equations by completing the square
  • Introducing imaginary solutions
  • Simplifying complex solutions with or without radicals

What to Watch Out For

Don't get too caught up in your work and neglect the following:

  • Applying the square root rule only when you have ax2 = c
  • Using the correct signs when applying the multiplication property of 0
  • Watching the order of operations when simplifying the work in the quadratic formula
  • Simplifying the fraction correctly in the quadratic formula
  • Pulling out the square root of −1 when determining imaginary roots

Applying the Square Root Rule to Quadratic Equations

606–613 Solve each quadratic equation using the square root rule.

606. x2 = 25

607. x2 = 121

608. 3y2 = 27

609. 5z2 = 80

610. n2 − 100 = 0

611. m2 − 1 = 0

612. 4x2 − 9 = 0

613. 24x2 − 150 = 0

Solving Quadratic Equations Using Factoring

614–629 Solve the quadratic equations using factoring.

614. x2 − 2x − 15 = 0

615. y2 + 15y + 44 = 0

616. 2x2 + x − 6 = 0

617. 3x2 − 8x + 5 = 0

618. y2 − 3y = 0

619. z2 = 7z

620. 2x2 + x = 0

621. 3y2 = 2y

622. 8x2 − 6x − 9 = 0

623. 10x2 + 29x + 10 = 0

624. 16x2 + 4x − 2 = 0

625. 6x2 − 9x − 15 = 0

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Applying the Quadratic Formula to Quadratic Equations

630–641 Solve each quadratic equation using the quadratic formula.

630. x2 + 3x − 4 = 0

631. x2 − 8x + 12 = 0

632. 2x2 + x − 6 = 0

633. 10x2 + 13x + 4 = 0

634. x2 − 3x − 1 = 0

635. x2 + 5x + 2 = 0

636. 2x2 − x − 5 = 0

637. 2x2 − 4x − 5 = 0

638. 3x2 + 6x + 1 = 0

639. x2 − 7x − 17 = 0

640. 2x2 + 8x + 3 = 0

641. x2 − 12x + 9 = 0

Completing the Square to Solve Quadratic Equations

642–645 Solve each quadratic equation by “completing the square.”

642. x2 + 2x − 24 = 0

643. 2x2 + 11x − 40 = 0

644. x2 − 4x + 2 = 0

645. x2 − 12x − 9 = 0

Writing Complex Numbers in the Standard a + bi Form

646–653 Rewrite each as a complex number in the form a + bi.

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Finding Complex Solutions Using the Quadratic Formula

654–655 Use the quadratic formula to solve the equations. Write your answers as complex numbers.

654. x2 + 4x + 8 = 0

655. x2 + x + 25 = 0

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