For what values of k does x²+2kx+7k-12=0 have equal roots

Question

For what values of $k$ does the equation $x^2 + 2kx + 7k - 12 = 0$ have equal roots?

Solution — Step by Step

A quadratic equation $ax^2 + bx + c = 0$ has equal (repeated) roots when its discriminant is zero:

\Delta = b^2 - 4ac = 0

Equal roots means both roots are the same: $x = -b/(2a)$ .

This is the condition because the quadratic formula gives $x = \frac{-b \pm \sqrt{\Delta}}{2a}$ . When $\Delta = 0$ , the $\pm\sqrt{0} = 0$ term vanishes and both roots are identical.

From $x^2 + 2kx + (7k - 12) = 0$ :

$a = 1$
$b = 2k$
$c = 7k - 12$

\Delta = b^2 - 4ac = 0

(2k)^2 - 4(1)(7k - 12) = 0

4k^2 - 28k + 48 = 0

Divide by 4:

k^2 - 7k + 12 = 0

Factorise: $k^2 - 7k + 12 = (k - 3)(k - 4) = 0$

k = 3 \quad \text{or} \quad k = 4

\boxed{k = 3 \text{ or } k = 4}

For $k = 3$ : Equation is $x^2 + 6x + 9 = 0 = (x+3)^2 = 0$ → $x = -3$ (double root) ✓

For $k = 4$ : Equation is $x^2 + 8x + 16 = 0 = (x+4)^2 = 0$ → $x = -4$ (double root) ✓

Both give perfect square quadratics, confirming equal roots.

Why This Works

The discriminant $\Delta = b^2 - 4ac$ tells us the nature of roots:

$\Delta > 0$ : two distinct real roots
$\Delta = 0$ : two equal (real) roots (one repeated root)
$\Delta < 0$ : no real roots (complex conjugate pair)

When we set $\Delta = 0$ and solve for the parameter $k$ , we get the values of $k$ that make the quadratic a perfect square (of the form $(x + p)^2 = 0$ ).

The elegant check: for $k = 3$ , the quadratic $x^2 + 6x + 9 = (x+3)^2$ , and for $k = 4$ , it’s $x^2 + 8x + 16 = (x+4)^2$ . Both are perfect squares, confirming our answer.

Alternative Method

Using completing the square: if $x^2 + 2kx + (7k-12) = 0$ has equal roots, it must be a perfect square of the form $(x + k)^2 = x^2 + 2kx + k^2 = 0$ .

Comparing: we need $c = k^2$ , i.e., $7k - 12 = k^2$ , i.e., $k^2 - 7k + 12 = 0$ . Same equation as before.

Common Mistake

Students sometimes confuse “equal roots” with “real roots.” Equal roots is a more specific condition: not only must the roots be real, they must both be the same value ( $\Delta = 0$ ). For just real roots, we need $\Delta \geq 0$ . For complex roots, $\Delta < 0$ . The question asks for equal roots — set $\Delta = 0$ exactly.

This type of problem (find parameter for given nature of roots) is standard in CBSE Class 10 and JEE. The technique is always: (1) write out $a$ , $b$ , $c$ in terms of the parameter; (2) apply the discriminant condition; (3) solve the resulting equation for the parameter; (4) verify by substitution. These four steps cover all variants of this question type.

Question

Solution — Step by Step

Why This Works

Alternative Method

Common Mistake

Try These Next