Solve the following quadratic equation by factoring out the GCF: 52x^2+6x=0 2 x 2x

To solve the quadratic equation \(52x^2 + 6x = 0\) by factoring out the greatest common factor (GCF), follow these steps:<br /><br />1. **Identify the GCF of the terms**:<br /> The terms in the equation are \(52x^2\) and \(6x\). The GCF of the coefficients \(52\) and \(6\) is \(2\), and the GCF of the variable part \(x^2\) and \(x\) is \(x\). Therefore, the GCF of the entire expression is \(2x\).<br /><br />2. **Factor out the GCF**:<br /> Factor \(2x\) out of each term in the equation:<br /> \[<br /> 52x^2 + 6x = 2x(26x) + 2x(3)<br /> \]<br /> Simplify the expression inside the parentheses:<br /> \[<br /> 52x^2 + 6x = 2x(26x + 3)<br /> \]<br /><br />3. **Set each factor equal to zero**:<br /> The factored form of the equation is \(2x(26x + 3) = 0\). Set each factor equal to zero and solve for \(x\):<br /> \[<br /> 2x = 0 \quad \text{or} \quad 26x + 3 = 0<br /> \]<br /><br />4. **Solve each equation**:<br /> For \(2x = 0\):<br /> \[<br /> x = 0<br /> \]<br /> For \(26x + 3 = 0\):<br /> \[<br /> 26x + 3 = 0<br /> \]<br /> Subtract 3 from both sides:<br /> \[<br /> 26x = -3<br /> \]<br /> Divide both sides by 26:<br /> \[<br /> x = -\frac{3}{26}<br /> \]<br /><br />So, the solutions to the quadratic equation \(52x^2 + 6x = 0\) are:<br />\[<br />x = 0 \quad \text{or} \quad x = -\frac{3}{26}<br />\]