Question #245447

solve the following quadratic equations. Leave surds in your answers.

a) (x-3)^2-3=0

b) (3x-7)^2 =8


2a) x^4-5x^2+4=0

b) x-8=2squareroot x


1
Expert's answer
2021-10-04T12:17:58-0400

a) (x3)23=0(x-3)^{2}-3=0

Add 33 to both sides:

(x3)23+3=0+3(x-3)^{2}-3+\textcolor{primary}{3}=0+\textcolor{primary}{3}

Add like terms:

(x3)2=3(x-3)^{2}=3

Take the square root for both sides:

x3=±3x-3=\pm \sqrt{3}

Add 33 to both sides:

x=3±3x=3\pm \sqrt{3}

The solutions are x=33x=3- \sqrt{3} and x=3+3x=3+\sqrt{3}

----------------------------------------------------------------------------------------------------------------------------------------

b)(3x7)2=8(3x-7)^{2}=8

Take the square root for both sides:

3x7=±83x-7=\pm \sqrt{8}

Add 77 to both sides:

3x=7±83x=7\pm \sqrt{8} ​

Divide both sides by 33:

x=7±83x=\frac{7\pm\sqrt{8}}{3}

Simplify the expression 8\sqrt{8} as 222\sqrt{2} :

x=7±223x=\frac{7\pm2\sqrt{2}}{3}

----------------------------------------------------------------------------------------------------------------------------------------

2a)x45x2+4=0x^4-5x^2+4=0

Find two numbers whose product is 44 "the third term" and have the sum of 5-5 "coefficient of the second term":

The numbers are 4-4 and 1-1, such that 4×1=4-4 \times -1 =4 and 4+(1)=5-4+(-1)=-5

Now factor the expression:

(x21)(x24)=0(x^2-1)(x^2-4)=0

Using Zero-Product Property, it follows that:

(x21)=0(x^2-1)=0 or (x24)=0(x^2-4)=0

Take the first equation:

x21=0x^2-1=0

Add 11 to both sides:

x2=1x^2=1

Take the square root for both sides:

x=±1x=\pm1

Take the second equation:

x24=0x^2-4=0

Add 44 to both sides:

x2=4x^2=4

Take the square root for both sides:

x=±2x=\pm2

Therefore, the solutions are 1,1,2,and 2-1,1,-2, \text{and}~ 2

----------------------------------------------------------------------------------------------------------------------------------------

b)x8=2xx-8=2\sqrt{x}

Square both sides:

(x8)2=(2x)2(x-8)^2=(2\sqrt{x})^2

Expand the square of the binomial on the left-hand side:

x216x+64=(2x)2x^2-16x+64=(2\sqrt{x})^2

Square the right-hand side:

x216x+64=4xx^2-16x+64=4x

Subtract 4x4x from both sides:

x216x+644x=4x4xx^2-16x+64-4x=4x-4x

Add like terms:

x220x+64=0x^2-20x+64=0

Write 20x-20x as a difference:

x24x16x+64=0x^2-4x-16x+64=0

Factor out xx from the first two expressions:

x(x4)16x+64=0x(x-4)-16x+64=0

Factor out 16-16 from the second two expressions:

x(x4)16(x(x-4)-16( x4)x-4) =0=0

Factor out (x4)(x-4) from the expression:

(x4)(x16)=0(x-4)(x-16)=0

Using Zero-Product Property, it follows that:

(x4)=0  or  (x16)=0(x-4)=0~~\text{or}~~(x-16)=0

x=4  or  x=16x=4~~\text{or}~~x=16


Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!
LATEST TUTORIALS
APPROVED BY CLIENTS