Chapter 6 - Radical Functions and Rational Exponents - 6-5 Solving Square Root and Other Radical Equations - Practice and Problem-Solving Exercises - Page 397: 90

$x = -\frac{3}{4}, -2$

To factor a quadratic equation in the form $ax^2 + bx + c = 0$, we look at factors of the product of $a$ and $c$ such that, when added together, equal $b$. For the equation $4x^2 + 11x + 6 = 0$, $(a)(c)$ is $(4)(6)$, or $24$, but when added together will equal $b$ or $11$. Both factors need to be positive, in this case. We came up with the following possibilities: $(a)(c)$ = $(24)(1)$ $b = 25$ $(a)(c)$ = $(12)(2)$ $b = 14$ $(a)(c)$ = $(8)(3)$ $b = 11$ The third pair works. We will use that pair to split the middle term: $4x^2 + 8x + 3x + 6 = 0$ Now, we can factor by grouping. We group the first two terms together and the last two terms together: $(4x^2 + 8x) + (3x + 6) = 0$ We see that $4x$ is a common factor for the first group and $3$ is a common factor for the second group, so let us factor them out: $4x(x + 2) + 3(x + 2) = 0$ We see that $x + 2$ is common to both groups, so we put that binomial in parentheses. The other binomial will be $4x + 3$, which is composed of the coefficients sitting in front of the binomials. We now have the two factors: $(4x + 3)(x + 2) = 0$ We now set each factor equal to zero and solve: $4x + 3 = 0$ Subtract $4$ from each side of the equation to isolate constants to the right side of the equation: $4x = -3$ Divide each side by $4$ to solve for $x$: $x = -\frac{3}{4}$ Let's set the other factor equal to zero: $x + 2 = 0$ Subtract $2$ from each side of the equation to solve for $x$: $x = -2$ The solution is $x = -\frac{3}{4}, -2$. To check if our solutions are correct, we plug our solutions back into the equation to see if the left and right sides equal one another. Let's plug in $x = -\frac{3}{4}$ first: $4(-\frac{3}{4})^2 + 11(-\frac{3}{4}) + 6 = 0$ Evaluate the exponent first: $4(\frac{9}{16}) + 11(-\frac{3}{4}) + 6 = 0$ Multiply next, according to order of operations: $\frac{36}{16} - \frac{33}{4} + 6 = 0$ Convert the fractions so that they have the same denominator, which is $16$, in this case: $\frac{36}{16} - \frac{132}{16} + 6 = 0$ Subtract the fractions: $-\frac{96}{16} + 6 = 0$ Simplify the fraction: $-6 + 6 = 0$ Add to simplify: $0 = 0$ The left and right sides are equal; therefore, this solution is correct. Let's check $x = -2$: $4(-2)^2 + 11(-2) + 6 = 0$ Evaluate the exponent first: $4(4) + 11(-2) + 6 = 0$ Multiply next, according to order of operations: $16 - 22 + 6 = 0$ Subtract or add from left to right, according to order of operations: $-6 + 6 = 0$ Add to simplify: $0 = 0$ The left and right sides are equal; therefore, this solution is correct.