Factor the expression completely. 25x^2-70x-15 25x^2-70x-15=square

To factor the expression \(25x^2 - 70x - 15\) completely, we can follow these steps:<br /><br />1. **Look for a common factor:** The greatest common factor of the coefficients 25, 70, and 15 is 5. Factor out 5 from the entire expression:<br /> \[<br /> 25x^2 - 70x - 15 = 5(5x^2 - 14x - 3)<br /> \]<br /><br />2. **Factor the quadratic inside the parentheses:** Now, we need to factor the quadratic \(5x^2 - 14x - 3\).<br /><br /> We look for two numbers that multiply to \(5 \times (-3) = -15\) and add to \(-14\). These numbers are \(-15\) and \(1\).<br /><br />3. **Rewrite the middle term using these numbers:**<br /> \[<br /> 5x^2 - 14x - 3 = 5x^2 - 15x + x - 3<br /> \]<br /><br />4. **Group terms and factor by grouping:**<br /> \[<br /> (5x^2 - 15x) + (x - 3)<br /> \]<br /> Factor out the common factors in each group:<br /> \[<br /> 5x(x - 3) + 1(x - 3)<br /> \]<br /><br />5. **Factor out the common binomial factor:**<br /> \[<br /> (5x + 1)(x - 3)<br /> \]<br /><br />6. **Combine with the factor we initially factored out:**<br /> \[<br /> 5(5x + 1)(x - 3)<br /> \]<br /><br />Thus, the completely factored form of the expression \(25x^2 - 70x - 15\) is:<br />\[<br />5(5x + 1)(x - 3)<br />\]