1. Equations and Inequalities

Completing the Square

1. Equations and inequalities

Completing the Square

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Multiple Choice

Solve the given quadratic equation by completing the square. $x^2+3x-5=0$

$x=-\frac32,x=\frac52$

$x=-\frac32,x=\sqrt{29}$

$x=\frac{-3+\sqrt{29}}{2},x=\frac{-3-\sqrt{29}}{2}$

$x=\frac{3+\sqrt{29}}{2},x=\frac{3-\sqrt{29}}{2}$

Verified step by step guidance

Start by rewriting the quadratic equation in the form $x^2 + 3x - 5 = 0$. To complete the square, we need to focus on the terms involving $x$.

Move the constant term to the other side of the equation: $x^2 + 3x = 5$.

To complete the square, take half of the coefficient of $x$, which is $\frac{3}{2}$, and square it to get $\left(\frac{3}{2}\right)^2 = \frac{9}{4}$. Add and subtract this value inside the equation: $x^2 + 3x + \frac{9}{4} = 5 + \frac{9}{4}$.

Rewrite the left side as a perfect square trinomial: $(x + \frac{3}{2})^2 = \frac{29}{4}$.

Solve for $x$ by taking the square root of both sides: $x + \frac{3}{2} = \pm \sqrt{\frac{29}{4}}$. Then, isolate $x$ to find the solutions: $x = -\frac{3}{2} \pm \frac{\sqrt{29}}{2}$.