Find the smaller value of x that satisfies the equation x$^2$ + 7x + 10 = 0

Analyze the question: We need to solve the quadratic equation x² + 7x + 10 = 0 and pick the smaller of its two roots.

Concept: To factorise x² + bx + c, find two numbers that multiply to give c and add to give b.

Solution:

Here, c = 10 and b = 7.

• Factors of 10: (1, 10), (2, 5)
• Which pair adds to 7? 2 + 5 = 7 ✓

So:

x² + 7x + 10 = 0 x² + 2x + 5x + 10 = 0 x(x + 2) + 5(x + 2) = 0 (x + 2)(x + 5) = 0

So x + 2 = 0, giving x = -2, or x + 5 = 0, giving x = -5.

Check:

• x = -2: 4 - 14 + 10 = 0 ✓
• x = -5: 25 - 35 + 10 = 0 ✓

The roots are -5 and -2. Since -5 is further left on the number line, -5 is the smaller value.

Tip: When the question asks for the "smaller" value, remember that with negative numbers, the one with the bigger digit is smaller (-5 < -2). Many students lose marks by picking -2 here. Also, for a quick check, the sum of roots should equal -b (here -7) and the product should equal c (here 10): (-5) + (-2) = -7 ✓ and (-5)(-2) = 10 ✓.