Question

Solve the quadratic equation $x^2 - 5x - 10 = 0$ and give your answer correct to two decimal places.

Hint:

For quadratic equations, use $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ and round final answers as required.

Answer 1

Concept: Use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] for equation $ax^2 + bx + c = 0$.
Step 1: Identify coefficients. Given: \[ x^2 - 5x - 10 = 0 \] \[ a = 1,\quad b = -5,\quad c = -10 \]
Step 2: Substitute into formula. \[ x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(1)(-10)}}{2(1)} \] \[ x = \frac{5 \pm \sqrt{25 + 40}}{2} \]
Step 3: Simplify. \[ x = \frac{5 \pm \sqrt{65}}{2} \]
Step 4: Approximate value. \[ \sqrt{65} \approx 8.062 \] \[ x_1 = \frac{5 + 8.062}{2} = \frac{13.062}{2} \approx 6.53 \] \[ x_2 = \frac{5 - 8.062}{2} = \frac{-3.062}{2} \approx -1.53 \]
Conclusion: The solutions correct to two decimal places are: \[ x = 6.53 \quad \text{or} \quad x = -1.53 \]