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Unit 15: Solving Quadratic Equations

The same quadratic equation can be solved four ways -- factoring, square roots, completing the square, or the quadratic formula -- and they agree.

6 topics · 20 questions

Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.

Solving Quadratics by Factoring

  1. How it worksHow does the zero product property turn a factored equation into two simple linear equations?
  2. Worked exampleSolve x2 - 5x + 6 = 0 by factoring.
  3. Worked exampleSolve x2 + 3x = 0 by factoring.
  4. Common mistakeWhy doesn't the zero product property work if a quadratic equation isn't set equal to zero first?
Your turn1 of 2
Solve x2 - (-4+7)x + -4·7 = 0 by factoring. Find the larger solution.

Solving Quadratics by Taking Square Roots

  1. How it worksHow does taking the square root of both sides of x2 = c produce a plus-or-minus pair of solutions?
  2. Worked exampleSolve the quantity x minus 3, squared, equals 25, by taking square roots.
  3. Common mistakeWhy doesn't taking the square root of both sides of x2 = 16 give you only positive 4?
Your turn1 of 2
Solve the quantity x minus 7, squared, equals 72, by taking square roots. Find the larger solution.

Completing the Square

  1. How it worksHow does adding and subtracting the same number let you rewrite x2 + 6x as a perfect square?
  2. Worked exampleSolve x2 + 6x - 7 = 0 by completing the square.
  3. Worked exampleFind the vertex form of y = x2 - 8x + 3 by completing the square.
  4. Common mistakeWhy can't you complete the square by adding a number to just one side of the equation?
Your turn1 of 2
Solve x2 - (-4+7)x + -4·7 = 0 by completing the square. Find the larger solution.

The Quadratic Formula

  1. How it worksHow does completing the square on the general equation ax2 + bx + c = 0 produce the quadratic formula?
  2. Worked exampleSolve 2x2 + 5x - 3 = 0 using the quadratic formula.
  3. Common mistakeWhy does the entire numerator, not just the square root part, need to be divided by 2a?
Your turn1 of 2
Solve 3x2 - 3·(-4+9)x + 3*-4·9 = 0 using the quadratic formula. Find the larger solution.

The Discriminant

  1. How it worksHow does the sign of b2 - 4ac decide whether a parabola crosses the x-axis twice, once, or not at all?
  2. Worked exampleFind the discriminant of 2x2 - 4x + 5 and state how many real solutions it has.
  3. Going furtherWhat happens to a parabola's graph as its discriminant shrinks from positive down through zero to negative?
Your turn1 of 2
Find the discriminant of 4x2 + 8x + 11.

Choosing a Method to Solve a Quadratic

  1. How it worksHow do you decide between factoring, square roots, and the quadratic formula just by looking at an equation's form?
  2. Worked exampleSolve x2 - 9 = 0 by choosing the fastest method available.
  3. Ties togetherHow does every method for solving a quadratic connect back to finding where its parabola crosses the x-axis?
Your turn1 of 2
Solve x2 - 72 = 0 by choosing the fastest method available. Find the positive solution.