Answer:
x = 4, -7
Step-by-step explanation:
1. <u>Solve the equation using the quadratic formula</u>
The <u>Quadratic formula</u> provides the solution for ![Ax^{2} +Bx+C=0](https://tex.z-dn.net/?f=Ax%5E%7B2%7D%20%2BBx%2BC%3D0)
in which A, B, and C are numbers (or coefficients), as follows:
![\frac{-b+\sqrt{b^{2}-4ac } }{2a}](https://tex.z-dn.net/?f=%5Cfrac%7B-b%2B%5Csqrt%7Bb%5E%7B2%7D-4ac%20%7D%20%7D%7B2a%7D)
2. <u>Determine the quadratic equation’s coefficients A, B, and C</u>
The <u>coefficients</u> of our equation,
, are:
A = 5
B = 15
C = -140
3. <u>Plug these coefficients into the quadratic formula</u>
<u />![\frac{-b+\sqrt{b^{2}-4ac } }{2a}=\frac{-15+\sqrt{15^{2}-4x5x-140 } }{2x5}](https://tex.z-dn.net/?f=%5Cfrac%7B-b%2B%5Csqrt%7Bb%5E%7B2%7D-4ac%20%7D%20%7D%7B2a%7D%3D%5Cfrac%7B-15%2B%5Csqrt%7B15%5E%7B2%7D-4x5x-140%20%7D%20%7D%7B2x5%7D)
Calculate the expression inside the parentheses.
Simplify exponents and square roots.
![\frac{-15+\sqrt{15^{2}-4x5x-140 } }{2x5}](https://tex.z-dn.net/?f=%5Cfrac%7B-15%2B%5Csqrt%7B15%5E%7B2%7D-4x5x-140%20%7D%20%7D%7B2x5%7D)
![\frac{-15+\sqrt{225-4x5x-140 } }{2x5}](https://tex.z-dn.net/?f=%5Cfrac%7B-15%2B%5Csqrt%7B225-4x5x-140%20%7D%20%7D%7B2x5%7D)
Perform any multiplication or division, from left to right.
![\frac{-15+\sqrt{225-20x-140 } }{2x5}](https://tex.z-dn.net/?f=%5Cfrac%7B-15%2B%5Csqrt%7B225-20x-140%20%7D%20%7D%7B2x5%7D)
![\frac{-15+\sqrt{225-2800 } }{2x5}](https://tex.z-dn.net/?f=%5Cfrac%7B-15%2B%5Csqrt%7B225-2800%20%7D%20%7D%7B2x5%7D)
![\frac{-15+\sqrt{3025} }{2x5}](https://tex.z-dn.net/?f=%5Cfrac%7B-15%2B%5Csqrt%7B3025%7D%20%7D%7B2x5%7D)
![\frac{-15+\sqrt{3025} }{10}](https://tex.z-dn.net/?f=%5Cfrac%7B-15%2B%5Csqrt%7B3025%7D%20%7D%7B10%7D)
to get the result:
![x=\frac{-15+\sqrt{3025} }{10}](https://tex.z-dn.net/?f=x%3D%5Cfrac%7B-15%2B%5Csqrt%7B3025%7D%20%7D%7B10%7D)
4. <u>Simplify square root</u>
Simplify 3025 by finding its prime factors.
3025
/ \
/ \
5 605
/ \
/ \
5 "<u>121</u>"
/ \
/ \
11 11
The prime factorization of 3025 is 5² • 11²
Write the prime factors.
= √5 • 5 • 11 • 11
Group the prime factors into pairs and rewrite in exponent form.
√5 • 5 • 11 • 11 = ![\sqrt{5^{2}x11^{2} }](https://tex.z-dn.net/?f=%5Csqrt%7B5%5E%7B2%7Dx11%5E%7B2%7D%20%20%7D)
Use rule
= x to simplify futher.
= 5 • 11
5 • 11 = 55
5. Solve the equation for x
x = ![\frac{-15+55}{10}](https://tex.z-dn.net/?f=%5Cfrac%7B-15%2B55%7D%7B10%7D)
The ± means two answers are possible:
x = 4
or
x = - 7