Find the Roots of the Quadratic Equation by Completing the Square: 2x² − 7x + 3 = 0
Question
Find the roots of the quadratic equation by the method of completing the square:
\[ 2x^2 – 7x + 3 = 0 \]Solution
\[
2x^2 – 7x = -3
\]
Divide both sides by 2:
\[
x^2 – \frac{7}{2}x = -\frac{3}{2}
\]
Add the square of half the coefficient of \(x\) to both sides:
\[
x^2 – \frac{7}{2}x + \left(\frac{7}{4}\right)^2
=
-\frac{3}{2} + \frac{49}{16}
\]
\[
\left(x-\frac{7}{4}\right)^2
=
\frac{-24+49}{16}
=
\frac{25}{16}
\]
Taking square roots:
\[
x-\frac{7}{4}
=
\pm \frac{5}{4}
\]
\[
x
=
\frac{7}{4}\pm\frac{5}{4}
\]
Hence,
\[
x=\frac{12}{4}=3
\]
or
\[
x=\frac{2}{4}=\frac{1}{2}
\]
Answer
\[
\boxed{x=3 \quad \text{or} \quad x=\frac{1}{2}}
\]