Factor completely. 288d^2-240d+50 square

To factor the quadratic expression \(288d^2 - 240d + 50\) completely, we can follow these steps:<br /><br />1. **Find the greatest common factor (GCF):**<br /> First, identify the GCF of the coefficients 288, 240, and 50.<br /> \[<br /> \text{GCF}(288, 240, 50) = 10<br /> \]<br /> So, we can factor out 10 from each term:<br /> \[<br /> 288d^2 - 240d + 50 = 10(28.8d^2 - 24d + 5)<br /> \]<br /><br />2. **Factor the quadratic expression inside the parentheses:**<br /> We need to factor \(28.8d^2 - 24d + 5\). To do this, we can use the quadratic formula to find the roots of the quadratic equation \(28.8d^2 - 24d + 5 = 0\).<br /><br /> The quadratic formula is given by:<br /> \[<br /> d = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}<br /> \]<br /> Here, \(a = 28.8\), \(b = -24\), and \(c = 5\).<br /><br /> Plugging in these values:<br /> \[<br /> d = \frac{-(-24) \pm \sqrt{(-24)^2 - 4 \cdot 28.8 \cdot 5}}{2 \cdot 28.8}<br /> \]<br /> \[<br /> d = \frac{24 \pm \sqrt{576 - 576}}{57.6}<br /> \]<br /> \[<br /> d = \frac{24 \pm \sqrt{0}}{57.6}<br /> \]<br /> \[<br /> d = \frac{24}{57.6}<br /> \]<br /> \[<br /> d = \frac{5}{12}<br /> \]<br /><br /> Since the discriminant (\(\Delta = b^2 - 4ac\)) is zero, the quadratic expression has a repeated root. Therefore, it can be factored as:<br /> \[<br /> 28.8d^2 - 24d + 5 = 28.8 \left(d - \frac{5}{12}\right)^2<br /> \]<br /><br />3. **Combine the factored form with the GCF:**<br /> Now, we combine this with the GCF we factored out initially:<br /> \[<br /> 288d^2 - 240d + 50 = 10 \cdot 28.8 \left(d - \frac{5}{12}\right)^2<br /> \]<br /><br />4. **Simplify the expression:**<br /> \[<br /> 10 \cdot 28.8 = 288<br /> \]<br /> So, the completely factored form is:<br /> \[<br /> 288d^2 - 240d + 50 = 288 \left(d - \frac{5}{12}\right)^2<br /> \]<br /><br />Therefore, the completely factored form of \(288d^2 - 240d + 50\) is:<br />\[<br />\boxed{288 \left(d - \frac{5}{12}\right)^2}<br />\]