Substitute x = 3: 9 - 3k + 6 = 0 → k = 5. Then find other root.

Find the Value of k If x = 3 Is a Root of x² - kx + 6 = 0

Question

Given that $x = 3$ is a root of the quadratic equation $x^2 - kx + 6 = 0$ , find the value of $k$ . Also find the other root.

Solution — Step by Step

Since $x = 3$ is a root, it must satisfy the equation. We plug in $x = 3$ directly:

3^2 - k(3) + 6 = 0

9 - 3k + 6 = 0

Simplify the left side:

15 - 3k = 0

3k = 15 \implies k = 5

So the equation becomes $x^2 - 5x + 6 = 0$ .

Now we factor $x^2 - 5x + 6 = 0$ . We need two numbers that multiply to $6$ and add to $-5$ . Those are $-2$ and $-3$ :

(x - 2)(x - 3) = 0

So $x = 2$ or $x = 3$ . The other root is $\mathbf{x = 2}$ .

A quick sanity check: sum of roots $= 2 + 3 = 5 = k$ ✓, and product of roots $= 2 \times 3 = 6$ ✓ (matches the constant term).

Final answer: $k = 5$ , and the other root is $2$ .

Why This Works

A root of a polynomial equation is a value of $x$ that makes the equation equal to zero. This is the definition — nothing more. So if someone tells us $x = 3$ is a root, we can treat it as a direct substitution to extract the unknown.

Once we find $k = 5$ , the equation is fully determined. From there, factoring is the cleanest route to the second root for a quadratic with small integer coefficients.

Vieta’s formulas give us a shortcut and a verification tool simultaneously. For $ax^2 + bx + c = 0$ , the sum of roots is $-b/a$ and the product is $c/a$ . Checking both protects you from silly arithmetic errors in the exam.

Alternative Method — Using Vieta’s Formulas Directly

Once we know $k = 5$ , we can skip factoring entirely.

For $x^2 - 5x + 6 = 0$ :

Sum of roots $= 5$ (coefficient of $x$ with sign flipped)
One root is $3$

So the other root $= 5 - 3 = \mathbf{2}$ .

In CBSE board exams, if the quadratic has a known root, use Vieta’s to find the other root in one line. It’s faster than factoring and looks clean in your answer sheet — examiners love it.

Common Mistake

The most common error is forgetting the sign when reading off the sum of roots. Students see $x^2 - 5x + 6$ and write “sum of roots $= -5$ ” because they copy the coefficient directly. Remember: for $x^2 + bx + c = 0$ , sum of roots $= -b$ , not $b$ . Here $b = -5$ , so sum $= -(-5) = 5$ . This appeared as a one-mark verification step in CBSE 2024, and many students lost the mark here.

Question

Solution — Step by Step

Why This Works

Alternative Method — Using Vieta’s Formulas Directly

Common Mistake

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