Solve the equation x$^2$ - 3x - 10 = 0

Analyze the question: We need the values of x that make x² - 3x - 10 = 0. This is a quadratic equation, and factorisation works well here.

Concept: For x² + bx + c = 0, find two numbers that:

• multiply to give c (the constant term)
• add up to give b (the coefficient of x)

Solution:

Here, c = -10 and b = -3.

Pairs of numbers that multiply to -10: (1, -10), (-1, 10), (2, -5), (-2, 5).

Which pair adds up to -3? 2 + (-5) = -3 ✔

So the factors are 2 and -5:

x² - 3x - 10 = x² + 2x - 5x - 10 = x(x + 2) - 5(x + 2) = (x + 2)(x - 5)

Set each factor to zero:

• x + 2 = 0 → x = -2
• x - 5 = 0 → x = 5

Check:

• x = 5: 25 - 15 - 10 = 0 ✔
• x = -2: 4 + 6 - 10 = 0 ✔

Tip: Check the signs first. If the constant term is negative, the two numbers have opposite signs, and the larger one takes the sign of the middle term. Here the middle term is -3, so the larger number (5) is negative in the pair (2, -5), and the roots are the opposites: x = 5 and x = -2. Always substitute your answers back to confirm.