How do you solve x^2 + 9x = -7 graphically and algebraically?

Aug 11, 2017

$x \approx - 0.86 \mathmr{and} - 8.14$

Explanation:

${x}^{2} + 9 x = - 7$

${x}^{2} + 9 x + 7 = 0$

Algebraic Solution:

This is a quadratic eqation of the form: $a {x}^{2} + b x + c = 0$

$x = \frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2 a}$

Hence, $x = \frac{- 9 \pm \sqrt{{9}^{2} - 4 \cdot 1 \cdot 7}}{2 \cdot 1}$

$= \frac{- 9 \pm \sqrt{81 - 28}}{2}$

$= \frac{- 9 \pm \sqrt{53}}{2}$

$\approx \frac{- 9 \pm 7.28}{2}$

$\approx - 4.5 \pm 3.64$

Hence, $x \approx - 0.86 \mathmr{and} - 8.14$

Graphical Solution:

To find the zeros of $f \left(x\right) = {x}^{2} + 9 x + 7$ we plot $f \left(x\right)$ and find the $x -$intercepts. As below:

graph{x^2+9x+7 [-28.87, 28.85, -14.44, 14.43]}

As can be seen, the $x -$intercepts of $f \left(x\right)$ are $\approx - 0.86 \mathmr{and} - 8.14$

Hence, this is the graphical solution to the given equation.

Aug 11, 2017

$x = \frac{- 9 \pm \sqrt{53}}{2}$

Explanation:

${x}^{2} + 9 x = - 7$
${x}^{2} + 9 x + 7 = 0$
$x = \frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2 a}$
$x = \frac{- \left(9\right) \pm \sqrt{{\left(9\right)}^{2} - 4 \left(1\right) \left(7\right)}}{2 \left(1\right)}$
$x = \frac{- 9 \pm \sqrt{53}}{2}$

Solving the equation graphically would be inefficient, since the algebraic method is required to use the graphical method. However, if you are given a graph with the exact value of the x-intercepts, the x-intercepts will be the solutions to the equation.