Solve the quadratic equation x² − 5x + 6 = 0 using the quadratic formula, show all steps

Mathematics

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Step-by-step solution

6 steps

To solve the quadratic equation x2−5x+6=0x^2 - 5x + 6 = 0 using the quadratic formula, follow these steps:

  1. Identify coefficients

    In the equation ax2+bx+c=0ax^2 + bx + c = 0, we have:

    • a=1a = 1
    • b=−5b = -5
    • c=6c = 6

  2. Quadratic formula

    The quadratic formula is given by:

    x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

  3. Calculate the discriminant

    First, calculate b2−4acb^2 - 4ac:

    b2−4ac=(−5)2−4(1)(6)=25−24=1b^2 - 4ac = (-5)^2 - 4(1)(6) = 25 - 24 = 1

  4. Substitute values into the formula

    x=−(−5)±12(1)=5±12x = \frac{-(-5) \pm \sqrt{1}}{2(1)} = \frac{5 \pm 1}{2}

  5. Calculate the two possible values for xx

    • For x1x_1: x1=5+12=62=3x_1 = \frac{5 + 1}{2} = \frac{6}{2} = 3
    • For x2x_2: x2=5−12=42=2x_2 = \frac{5 - 1}{2} = \frac{4}{2} = 2

  6. Final solutions

    The solutions to the equation x2−5x+6=0x^2 - 5x + 6 = 0 are:

    x=3andx=2x = 3 \quad \text{and} \quad x = 2
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