Quadratic Formula Math Example 3

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Example 3

hard
Solve 2x2−5x−3=02x^2 - 5x - 3 = 0 using the quadratic formula.

Solution

  1. 1
    Step 1: Identify a=2a = 2, b=−5b = -5, c=−3c = -3.
  2. 2
    Step 2: Compute the discriminant: b2−4ac=(−5)2−4(2)(−3)=25+24=49b^2 - 4ac = (-5)^2 - 4(2)(-3) = 25 + 24 = 49.
  3. 3
    Step 3: Apply the formula: x=−(−5)±492(2)=5±74x = \frac{-(-5) \pm \sqrt{49}}{2(2)} = \frac{5 \pm 7}{4}.
  4. 4
    Step 4: Two solutions: x=5+74=3x = \frac{5 + 7}{4} = 3 or x=5−74=−12x = \frac{5 - 7}{4} = -\frac{1}{2}.

Answer

x=3 or x=−12x = 3 \text{ or } x = -\frac{1}{2}
The quadratic formula x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} solves any quadratic. Be careful with signs — here b=−5b = -5, so −b=5-b = 5. A perfect square discriminant means the roots are rational.

About Quadratic Formula

A formula giving the exact solutions to any quadratic equation ax2+bx+c=0ax^2 + bx + c = 0 directly from its three coefficients.

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More Quadratic Formula Examples

Example 1 easy

Solve [formula] by factoring.

Example 2 medium

Solve [formula] using the quadratic formula.

Example 4 easy

Solve [formula].

Example 5 hard

Solve [formula] and describe the solutions.

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