Math  /  Algebra

QuestionQUESTIION 4  (2 + 2 = 4 Marks \text { (2 + } 2 \text { = } 4 \text { Marks }
Solve each of the following quadratic equations for xx using "Quadratic Formula". Leave your answer in surd (exact) form. (a). 3(x2)2+4x2=5\frac{3}{(x-2)^{2}}+\frac{4}{x-2}=5

Studdy Solution

STEP 1

1. The given equation is a quadratic equation in the variable u=x2u = x - 2.
2. The quadratic formula x=b±b24ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} can be used to solve for uu.
3. The final answer will be converted back to xx after solving for uu.

STEP 2

1. Introduce a substitution to simplify the equation.
2. Rewrite the equation in terms of the new variable.
3. Simplify the equation to standard quadratic form.
4. Apply the quadratic formula to solve for the new variable.
5. Substitute back to find xx.

STEP 3

Introduce the substitution u=x2 u = x - 2 to simplify the equation.
3u2+4u=5 \frac{3}{u^2} + \frac{4}{u} = 5

STEP 4

Rewrite the equation in terms of u u .
3u2+4u5=0 \frac{3}{u^2} + \frac{4}{u} - 5 = 0

STEP 5

Multiply through by u2 u^2 to clear the denominators.
3+4u5u2=0 3 + 4u - 5u^2 = 0

STEP 6

Rearrange the equation to standard quadratic form au2+bu+c=0 au^2 + bu + c = 0 .
5u2+4u+3=0 -5u^2 + 4u + 3 = 0

STEP 7

Apply the quadratic formula u=b±b24ac2a u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} where a=5 a = -5 , b=4 b = 4 , and c=3 c = 3 .
u=4±424(5)32(5) u = \frac{-4 \pm \sqrt{4^2 - 4 \cdot (-5) \cdot 3}}{2 \cdot (-5)}

STEP 8

Simplify under the square root.
u=4±16+6010 u = \frac{-4 \pm \sqrt{16 + 60}}{-10}
u=4±7610 u = \frac{-4 \pm \sqrt{76}}{-10}

STEP 9

Simplify further.
u=4±21910 u = \frac{-4 \pm 2\sqrt{19}}{-10}
u=2(2±19)10 u = \frac{-2(2 \pm \sqrt{19})}{-10}
u=2±195 u = \frac{2 \pm \sqrt{19}}{5}

STEP 10

Substitute back for u=x2 u = x - 2 .
x2=2±195 x - 2 = \frac{2 \pm \sqrt{19}}{5}

STEP 11

Solve for x x .
x=2+2±195 x = 2 + \frac{2 \pm \sqrt{19}}{5}

STEP 12

Simplify the expression.
x=10+2±195 x = \frac{10 + 2 \pm \sqrt{19}}{5}
x=12±195 x = \frac{12 \pm \sqrt{19}}{5}
Thus, the solutions for x x are:
x=12+195andx=12195 x = \frac{12 + \sqrt{19}}{5} \quad \text{and} \quad x = \frac{12 - \sqrt{19}}{5}

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord