Solve y^2+18y+24

$$y ^ { 2 } + 18 y + 24$$

Factor

$$\left(y-\left(-\sqrt{57}-9\right)\right)\left(y-\left(\sqrt{57}-9\right)\right)$$

Steps Using the Quadratic Formula

Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$y^{2}+18y+24=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$y=\frac{-18±\sqrt{18^{2}-4\times 24}}{2}$$

Square 18.

$$y=\frac{-18±\sqrt{324-4\times 24}}{2}$$

Multiply -4 times 24.

$$y=\frac{-18±\sqrt{324-96}}{2}$$

Add 324 to -96.

$$y=\frac{-18±\sqrt{228}}{2}$$

Take the square root of 228.

$$y=\frac{-18±2\sqrt{57}}{2}$$

Now solve the equation y=\frac{-18±2\sqrt{57}}{2} when ± is plus. Add -18 to 2\sqrt{57}.

$$y=\frac{2\sqrt{57}-18}{2}$$

Divide -18+2\sqrt{57} by 2.

$$y=\sqrt{57}-9$$

Now solve the equation y=\frac{-18±2\sqrt{57}}{2} when ± is minus. Subtract 2\sqrt{57} from -18.

$$y=\frac{-2\sqrt{57}-18}{2}$$

Divide -18-2\sqrt{57} by 2.

$$y=-\sqrt{57}-9$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -9+\sqrt{57} for x_{1} and -9-\sqrt{57} for x_{2}.

$$y^{2}+18y+24=\left(y-\left(\sqrt{57}-9\right)\right)\left(y-\left(-\sqrt{57}-9\right)\right)$$

Steps Using Direct Factoring Method

Quadratic equations such as this one can be solved by a new direct factoring method that does not require guess work. To use the direct factoring method, the equation must be in the form x^2+Bx+C=0.

$$x ^ 2 +18x +24 = 0$$

Let r and s be the factors for the quadratic equation such that x^2+Bx+C=(x−r)(x−s) where sum of factors (r+s)=−B and the product of factors rs = C

$$r + s = -18 rs = 24$$

Two numbers r and s sum up to -18 exactly when the average of the two numbers is \frac{1}{2}*-18 = -9. You can also see that the midpoint of r and s corresponds to the axis of symmetry of the parabola represented by the quadratic equation y=x^2+Bx+C. The values of r and s are equidistant from the center by an unknown quantity u. Express r and s with respect to variable u. <div style='padding: 8px'><img src='https://opalmath.azureedge.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>

$$r = -9 - u s = -9 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 24

$$(-9 - u) (-9 + u) = 24$$

Simplify by expanding (a -b) (a + b) = a^2 – b^2

$$81 - u^2 = 24$$

Simplify the expression by subtracting 81 on both sides

$$-u^2 = 24-81 = -57$$

Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u

$$u^2 = 57 u = \pm\sqrt{57} = \pm \sqrt{57}$$

The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.

$$r =-9 - \sqrt{57} = -16.550 s = -9 + \sqrt{57} = -1.450$$

Evaluate

$$y^{2}+18y+24$$