Solve 3r^2-6r-6=0

Question

Solve the quadratic equation

Solve using the quadratic formula

Solve by completing the square

Solve using the PQ formula

r_{1} = 1 - 3, r_{2} = 1 + 3

Alternative Form

r_{1} \approx - 0.732051, r_{2} \approx 2.732051

Evaluate

3 r^{2} - 6 r - 6 = 0

Substitute a= 3,b= - 6 and c= - 6 into the quadratic formula r = \frac{- b \pm b ^{2} - 4 a c}{2 a}

r = \frac{6 \pm ( - 6 ) ^{2} - 4 \times 3 ( - 6 )}{2 \times 3}

Simplify the expression

r = \frac{6 \pm ( - 6 ) ^{2} - 4 \times 3 ( - 6 )}{6}

Simplify the expression

More Steps

Evaluate

(- 6)^{2} - 4 \times 3 (- 6)

Multiply

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Multiply the terms

4 \times 3 (- 6)

Rewrite the expression

- 4 \times 3 \times 6

Multiply the terms

- 72

(- 6)^{2} - (- 72)

Rewrite the expression

6^{2} - (- 72)

If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses

6^{2} + 72

Evaluate the power

36 + 72

Add the numbers

108

r = \frac{6 \pm 108}{6}

Simplify the radical expression

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Evaluate

108

Write the expression as a product where the root of one of the factors can be evaluated

36 \times 3

Write the number in exponential form with the base of 6

6^{2} \times 3

The root of a product is equal to the product of the roots of each factor

6^{2} \times 3

Reduce the index of the radical and exponent with 2

63

r = \frac{6 \pm 6 3}{6}

Separate the equation into 2 possible cases

r = \frac{6 + 6 3}{6} r = \frac{6 - 6 3}{6}

Simplify the expression

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Evaluate

r = \frac{6 + 6 3}{6}

Divide the terms

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Evaluate

\frac{6 + 6 3}{6}

Rewrite the expression

\frac{6 ( 1 + 3 )}{6}

Reduce the fraction

1 + 3

r = 1 + 3

r = 1 + 3 r = \frac{6 - 6 3}{6}

Simplify the expression

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Evaluate

r = \frac{6 - 6 3}{6}

Divide the terms

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Evaluate

\frac{6 - 6 3}{6}

Rewrite the expression

\frac{6 ( 1 - 3 )}{6}

Reduce the fraction

1 - 3

r = 1 - 3

r = 1 + 3 r = 1 - 3

Solution

r_{1} = 1 - 3, r_{2} = 1 + 3

Alternative Form

r_{1} \approx - 0.732051, r_{2} \approx 2.732051

Show Solution

Rewrite the equation

8 x^{2} + 8 y^{2} = x^{4} + y^{4} + 4 + 2 x^{2} y^{2}

Evaluate

3 r^{2} - 6 r - 6 = 0

Rewrite the expression

3 r^{2} - 6 r = 6

Use substitution

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Evaluate

3 r^{2} - 6 r

To covert the equation to rectangular coordinates using conversion formulas,substitute x^{2} + y^{2} for r^{2}

3 (x^{2} + y^{2}) - 6 r

Simplify the expression

3 x^{2} + 3 y^{2} - 6 r

3 x^{2} + 3 y^{2} - 6 r = 6

Simplify the expression

- 6 r = - 3 x^{2} - 3 y^{2} + 6

Square both sides of the equation

(- 6 r)^{2} = (- 3 x^{2} - 3 y^{2} + 6)^{2}

Evaluate

36 r^{2} = (- 3 x^{2} - 3 y^{2} + 6)^{2}

To covert the equation to rectangular coordinates using conversion formulas,substitute x^{2} + y^{2} for r^{2}

36 (x^{2} + y^{2}) = (- 3 x^{2} - 3 y^{2} + 6)^{2}

Divide both sides of the equation by 9

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

Calculate

4 x^{2} + 4 y^{2} = x^{4} + y^{4} + 4 + 2 x^{2} y^{2} - 4 x^{2} - 4 y^{2}

Move the expression to the left side

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

Calculate

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Evaluate

4 x^{2} + 4 x^{2}

Collect like terms by calculating the sum or difference of their coefficients

(4 + 4) x^{2}

Add the numbers

8 x^{2}

8 x^{2} + 4 y^{2} = x^{4} + y^{4} + 4 + 2 x^{2} y^{2} - 4 y^{2}

Solution

More Steps

Evaluate

4 y^{2} + 4 y^{2}

Collect like terms by calculating the sum or difference of their coefficients

(4 + 4) y^{2}

Add the numbers

8 y^{2}

8 x^{2} + 8 y^{2} = x^{4} + y^{4} + 4 + 2 x^{2} y^{2}

Show Solution

Solve the quadratic equation

Rewrite the equation

Graph