Question

Solve the quadratic equation by using a graphic approach. Round your answer to the hundredths place.

Answer 1

Answer:

x = 3.24, x = -1.24

Step-by-step explanation:

The standard form for a quadratic equation is $ax^2+bx+c=0$. For your equation a = 1, b = -2, c = -4. The quadratic formula you will be using is $x=\frac{-b\pm \sqrt{b^{2} -4ac} }{2a}$.

Plug in a = 1, b = -2, and c = -4 into the formula.

$=\frac{-\left(-2\right)\pm \sqrt{\left(-2\right)^2-4\cdot \:1\cdot \left(-4\right)}}{2\cdot \:1}$

We'll do the top part first:

$\sqrt{\left(-2\right)^2-4\cdot \:1\cdot \left(-4\right)}$

Apply rule $- (-a) = a$

$=\sqrt{\left(-2\right)^2+4\cdot \:1\cdot \:4}$

Apply exponent rule $(-a)^{n} =a^n$ if $n$ is even

$(-2)^2=2^2$

$=\sqrt{2^2+4\cdot \:1\cdot \:4}$

Multiply the numbers

$=\sqrt{2^2+16}$

$2^2=4$

$=\sqrt{4+16}$

Add

$=\sqrt{20}$

The prime factorization of 20 is $2^2*5$

20 divides by 2. 20 = 10 * 2

$=2*10$

10 divides by 2. 10 = 5 * 2

$=2* \:2*5$

2 & 5 are prime numbers so you don't need to factor them anymore

$=2*2*5$

$=2^2*5$

$=\sqrt{2^2\cdot \:5}$

Apply radical rule $\sqrt[n]{ab} =\sqrt[n]{a} \sqrt[n]{b}$

$=\sqrt{5}\sqrt{2^2}$

Apply radical rule $\sqrt[n]{a^{n} } =a$; $\sqrt{2^2} =2$

$=2\sqrt{5}$

$=\frac{-\left(-2\right)\pm \:2\sqrt{5}}{2\cdot \:1}$

Because of the $\pm$ you have to separate the solutions so that one is positive and the other is negative.

$x=\frac{-\left(-2\right)+2\sqrt{5}}{2\cdot \:1},\:x=\frac{-\left(-2\right)-2\sqrt{5}}{2\cdot \:1}$

Positive x:

$\frac{-\left(-2\right)+2\sqrt{5}}{2\cdot \:1}$

Apply rule $-(-a)=a$

$=\frac{2+2\sqrt{5}}{2\cdot \:1}$

Multiply

$=\frac{2+2\sqrt{5}}{2}$

Factor $2+2\sqrt{5}$ and rewrite it as $=2\cdot \:1+2\sqrt{5}$. Factor out 2 because it is the common term. $=2\left(1+\sqrt{5}\right)$.

$=\frac{2\left(1+\sqrt{5}\right)}{2}$

Divide 2 by 2

$x=1+\sqrt{5}$ or $x=3.24$ (You'll probably have to use a calculator for the square root of 5)

^Repeating the process of positive x for negative x in order to get $x=1-\sqrt{5}$ or $x=-1.24$