1Cademy - Solving $$x^2 + 10x + 4 = 15$$ by Completing the Square

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How to Solve a Quadratic Equation by Completing the Square

Example

Solving $x^2 + 10x + 4 = 15$ by Completing the Square

Solve $x^2 + 10x + 4 = 15$ by completing the square, demonstrating the procedure when constants appear on both sides of the equation.

Step 1 — Isolate the variable terms. The variable terms $x^2$ and $10x$ are already on the left, but the constant 4 must be moved to the right. Subtract 4 from both sides: $x^2 + 10x = 11$

Step 2 — Find $\left(\frac{1}{2} \cdot b\right)^2$ and add it to both sides. The coefficient of $x$ is 10, so $b = 10$ . Compute: $\left(\frac{1}{2}(10)\right)^2 = 5^2 = 25$ . Add 25 to both sides: $x^2 + 10x + 25 = 11 + 25$

Step 3 — Factor the perfect square trinomial. The left side factors as a binomial square: $(x + 5)^2 = 36$

Step 4 — Apply the Square Root Property: $x + 5 = \pm\sqrt{36}$

Step 5 — Simplify the radical and solve. Since 36 is a perfect square ( $6^2 = 36$ ): $x + 5 = \pm 6$ Write as two equations and solve each: $x + 5 = 6 \implies x = 1$ $x + 5 = -6 \implies x = -11$

Step 6 — Check both solutions: For $x = 1$ : $1^2 + 10(1) + 4 = 1 + 10 + 4 = 15$ ✓ For $x = -11$ : $(-11)^2 + 10(-11) + 4 = 121 - 110 + 4 = 15$ ✓

The solutions are $x = 1$ and $x = -11$ .

Unlike earlier examples where the equation was already in the form $x^2 + bx = c$ , this equation has a constant on the same side as the variable terms. Step 1 requires subtracting that constant from both sides before the completing-the-square process can begin. The trinomial $x^2 + 10x + 25$ matches the Binomial Squares Pattern because $(x + 5)^2 = x^2 + 10x + 25$ .

Updated 2026-06-17

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OpenStax Elementary Algebra

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