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Solve the equation: (3x + 6)² - 3(2x - 6) = 5x² + 4


NONSENSE WILL BE REPORTED​

Sagot :

Charmayanne

Question:

  • Solve the equation: (3x + 6)² - 3(2x - 6) = 5x² + 4

Answer:

  • [tex]\underline{\boxed{\sf{\pink{-\frac{5}{2}}}}}[/tex]

Solution:

Firstly, we should simplify the equation

[tex]\: (3x + 6)^2 - 3(2x - 6) = 5x^2 + 4[/tex]

[tex]\: 9x^2 + 36x + 36 - 6x + 18 - 5x^2 - 4 = 0[/tex]

[tex]\: 4x^2 + 30x + 50 = 0[/tex]

The square equation has the form:

  • [tex]\underline{\boxed{\pink{ax^2 + bx + c}}}[/tex]

In the above equation, these variables are:

[tex]\: a = 4, \: b = 30, \: c = 50[/tex]

Firstly, let's calculate delta

[tex]\: \Delta = b^2 - 4 \times a \times c[/tex]

[tex]\: \Delta = 30 \times 30 - 4 \times 4 \times 50 = 100[/tex]

Delta is greater than zero, so there are 2 solutions. Calculate them from the formulas:

[tex]\: x_1 = \frac{- b - \sqrt{\Delta}}{2 \times a} = \frac{-30 - \sqrt{100}}{2 \times 4} = -5[/tex]

[tex]\: x_2 = \frac{- b + \sqrt{\Delta}}{2 \times a} = \frac{- 30 + \sqrt{100}}{2 \times 4} = -\frac{5}{2}[/tex]

[tex]──────────────────────[/tex]

Xiaah

PROBLEM:

  • (3x + 6)² - 3(2x - 6) = 5x² + 4

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ANSWER:

  • [tex]{\large{\underline{\boxed{\sf{\red{x=\frac{-5}{2} }}}}}}\\{\large{\underline{\boxed{\sf{\red{x=-5 }}}}}}[/tex]

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SOLUTION:

  • So in this problem, we can use Factoring, Quadratic formula, and Completing the Square. But we'll gonna use Factoring.

[tex]\small{ \tt \purple{(3x^2+6)^2-3(2x-6)=5x^2+4}}[/tex]

[tex]\small{ \tt \purple{9x^2+36+36-3(2x-6)=5x^2+4}}[/tex]

[tex]\small{ \tt \purple{9x^2+36+36-6x+18=5x^2+4}}}}[/tex]

[tex]\small{ \tt \purple{9x^2+30+36+18=5x^2+4}}[/tex]

[tex]\small{ \tt \purple{9x^2+30+54=5x^2+4}}[/tex]

[tex]\small{ \tt \purple{9x^2+30+54-5x^2=4}}[/tex]

[tex]\small{ \tt \purple{4x^2+30x+54-4}}[/tex]

[tex]\small{ \tt \purple{4x^2+30x+54-4=0}}[/tex]

[tex]\small{ \tt \purple{4x^2+30x+50=0}}[/tex]

[tex]\small{ \tt \purple{2x^2+15x+25=0}}[/tex]

[tex]\small{ \tt \purple{a+b=15}}\\\small{ \tt \purple{ab=2\times25=50}}\\[/tex]

  • Since ab is positive, a and b have the same sign. Since a + b is positive, a and b are both positive. List all such integer pairs that give that product 50.

[tex]\small{ \tt \purple{1,50}}\\\small{ \tt \purple{2,25}}\\\small{ \tt \purple{5,10}}\\[/tex]

  • Calculate the sum for each pair.

[tex]\small{ \tt \purple{1+50=51}}\\\small{ \tt \purple{2+25=27}}\\\small{ \tt \purple{5+10=15}}\\[/tex]

  • The solution is the pair that gives a sum of 15.

[tex]\small{ \tt \purple{a=5}}\\\small{ \tt \purple{b=10}}\\[/tex]

  • Rewrite [tex]2x ^2 +15x+25[/tex] as [tex](2x^2+5x)+(10x+25).[/tex]

[tex]\small{ \tt \purple{(2x^2+5x)+(10x+25)}}\\[/tex]

  • Factor out x in the first and 5 in the second group.

[tex]\small{ \tt \purple{x(2x+5)+5(2x+5)}}\\[/tex]

  • Factor out the common term 2x + 5 by using the distributive property.

[tex]\small{ \tt \purple{(2x+5)(x+5)}}\\[/tex]

  • To find equation solutions, solve 2x + 5 = 0 and x + 5 = 0.

[tex]\small{ \tt \purple{Answer:}}\\{\large{\underline{\boxed{\sf{\red{x=\frac{-5}{2} }}}}}}\\{\large{\underline{\boxed{\sf{\red{x=-5 }}}}}}[/tex]

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