Solve r^2-4r-91=7

Question

Solve the quadratic equation

Solve using the quadratic formula

Solve by completing the square

Solve using the PQ formula

r_{1} = 2 - 102, r_{2} = 2 + 102

Alternative Form

r_{1} \approx - 8.099505, r_{2} \approx 12.099505

Evaluate

r^{2} - 4 r - 91 = 7

Move the expression to the left side

r^{2} - 4 r - 98 = 0

Substitute a= 1,b= - 4 and c= - 98 into the quadratic formula r = \frac{- b \pm b ^{2} - 4 a c}{2 a}

r = \frac{4 \pm ( - 4 ) ^{2} - 4 ( - 98 )}{2}

Simplify the expression

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Evaluate

(- 4)^{2} - 4 (- 98)

Multiply the numbers

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Evaluate

4 (- 98)

Multiplying or dividing an odd number of negative terms equals a negative

- 4 \times 98

Multiply the numbers

- 392

(- 4)^{2} - (- 392)

Rewrite the expression

4^{2} - (- 392)

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

4^{2} + 392

Evaluate the power

16 + 392

Add the numbers

408

r = \frac{4 \pm 408}{2}

Simplify the radical expression

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Evaluate

408

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

4 \times 102

Write the number in exponential form with the base of 2

2^{2} \times 102

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

2^{2} \times 102

Reduce the index of the radical and exponent with 2

2102

r = \frac{4 \pm 2 102}{2}

Separate the equation into 2 possible cases

r = \frac{4 + 2 102}{2} r = \frac{4 - 2 102}{2}

Simplify the expression

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Evaluate

r = \frac{4 + 2 102}{2}

Divide the terms

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Evaluate

\frac{4 + 2 102}{2}

Rewrite the expression

\frac{2 ( 2 + 102 )}{2}

Reduce the fraction

2 + 102

r = 2 + 102

r = 2 + 102 r = \frac{4 - 2 102}{2}

Simplify the expression

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Evaluate

r = \frac{4 - 2 102}{2}

Divide the terms

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Evaluate

\frac{4 - 2 102}{2}

Rewrite the expression

\frac{2 ( 2 - 102 )}{2}

Reduce the fraction

2 - 102

r = 2 - 102

r = 2 + 102 r = 2 - 102

Solution

r_{1} = 2 - 102, r_{2} = 2 + 102

Alternative Form

r_{1} \approx - 8.099505, r_{2} \approx 12.099505

Show Solution

Rewrite the equation

212 x^{2} + 212 y^{2} = x^{4} + y^{4} + 9604 + 2 x^{2} y^{2}

Evaluate

r^{2} - 4 r - 91 = 7

Rewrite the expression

r^{2} - 4 r = 91 + 7

Simplify the expression

r^{2} - 4 r = 98

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

x^{2} + y^{2} - 4 r = 98

Simplify the expression

- 4 r = - x^{2} - y^{2} + 98

Square both sides of the equation

(- 4 r)^{2} = (- x^{2} - y^{2} + 98)^{2}

Evaluate

16 r^{2} = (- x^{2} - y^{2} + 98)^{2}

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

16 (x^{2} + y^{2}) = (- x^{2} - y^{2} + 98)^{2}

Calculate

16 x^{2} + 16 y^{2} = x^{4} + y^{4} + 9604 + 2 x^{2} y^{2} - 196 x^{2} - 196 y^{2}

Move the expression to the left side

16 x^{2} + 16 y^{2} - (- 196 x^{2} - 196 y^{2}) = x^{4} + y^{4} + 9604 + 2 x^{2} y^{2}

Calculate

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Evaluate

16 x^{2} + 196 x^{2}

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

(16 + 196) x^{2}

Add the numbers

212 x^{2}

212 x^{2} + 16 y^{2} = x^{4} + y^{4} + 9604 + 2 x^{2} y^{2} - 196 y^{2}

Solution

More Steps

Evaluate

16 y^{2} + 196 y^{2}

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

(16 + 196) y^{2}

Add the numbers

212 y^{2}

212 x^{2} + 212 y^{2} = x^{4} + y^{4} + 9604 + 2 x^{2} y^{2}

Show Solution

Solve the quadratic equation

Rewrite the equation

Graph