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Find one value of \( x \) that is a solution to the equation: \( \begin{array}{l}(2 x+3)^{2}-6 x-9=0 \\ x=\end{array} \)

Ask by Brewer Summers. in the United States
Feb 22,2025

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One solution is \( x = -\frac{3}{2} \).

Solución

Solve the equation \( (2x+3)^2-6x-9=0 \). Solve the quadratic equation by following steps: - step0: Solve by factoring: \(\left(2x+3\right)^{2}-6x-9=0\) - step1: Expand the expression: \(4x^{2}+6x=0\) - step2: Factor the expression: \(2x\left(2x+3\right)=0\) - step3: Separate into possible cases: \(\begin{align}&2x=0\\&2x+3=0\end{align}\) - step4: Solve the equation: \(\begin{align}&x=0\\&x=-\frac{3}{2}\end{align}\) - step5: Rewrite: \(x_{1}=-\frac{3}{2},x_{2}=0\) The solutions to the equation \( (2x+3)^2-6x-9=0 \) are \( x = -\frac{3}{2} \) and \( x = 0 \). Therefore, one value of \( x \) that is a solution to the equation is \( x = -\frac{3}{2} \).

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Bonus Knowledge

To solve the equation \((2x+3)^{2}-6x-9=0\), we first expand and simplify the left side. 1. Expand the squared term: \((2x+3)^{2} = 4x^{2} + 12x + 9\). 2. Substitute it back into the equation: \(4x^{2} + 12x + 9 - 6x - 9 = 0\). 3. Simplify the equation: \(4x^{2} + (12x - 6x) + (9 - 9) = 0\), \(4x^{2} + 6x = 0\). 4. Factor out the common term: \(2x(2x + 3) = 0\). 5. Set each factor to zero: \(2x = 0 \implies x = 0\), \(2x + 3 = 0 \implies 2x = -3 \implies x = -\frac{3}{2}\). Thus, one value of \( x \) that is a solution to the equation is \( x = 0 \).

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