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3 years ago

Complete the steps in solving the quadratic function 7x – 9 = 7x2 – 49x by completing the square

Mathematics

1 answer:

siniylev [52]3 years ago

4 0

Answer:

x={-sqrt(103/7)+4, sqrt(103/7) +4}

Step-by-step explanation:

Subtract 7x from both sides: 7x^2-56x=-9

Divide both sides by 7: x^2-8x=-9/7

Solve for a (2ax=-8x): -4

Plug a back in: -9/7+(-4)^2=103/7

Complete the square: (x-4)^2=103/7

Therefore, x={-sqrt(103/7)+4, sqrt(103/7) +4}

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3 years ago

Convert the Cartesian equation (x 2 + y 2)2 = 4(x 2 - y 2) to a polar equation.

SashulF [63]

<u>ANSWER</u>

${r}^{2} = 4 \cos2\theta$

<u>EXPLANATION</u>

The Cartesian equation is

${( {x}^{2} + {y}^{2} )}^{2} = 4( {x}^{2} - {y}^{2} )$

We substitute

$x = r \cos( \theta)$

$y = r \sin( \theta)$

and

${x}^{2} + {y}^{2} = {r}^{2}$

This implies that

${( {r}^{2} )}^{2} = 4(( { r \cos\theta) }^{2} - {(r \sin\theta) }^{2} )$

Let us evaluate the exponents to get:

${r}^{4} = 4({ {r}^{2} \cos^{2}\theta } - {r}^{2} \sin^{2}\theta)$

Factor the RHS to get:

${r}^{4} = 4{r}^{2} ({ \cos^{2}\theta } - \sin^{2}\theta)$

Divide through by r²

${r}^{2} = 4 ({ \cos^{2}\theta } - \sin^{2}\theta)$

Apply the double angle identity

$\cos^{2}\theta -\sin^{2}\theta= \cos(2 \theta)$

The polar equation then becomes:

${r}^{2} = 4 \cos2\theta$

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