Using the discriminant, give the nature of the roots of $7 x^{3} + x^{2} - 35 x = 5$ $7x^3+x^2-35x=5$ ?

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Using the discriminant, give the nature of the roots of $7 x^{3} + x^{2} - 35 x = 5$ $7x^3+x^2-35x=5$ ?

Using the discriminant, give the nature of the roots of $7 x^{3} + x^{2} - 35 x = 5$ $7x^3+x^2-35x=5$ .
Also solve the equation.

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Answer 1 · 2017-03-13T13:11:39

We have real roots.

Explanation:

The discriminant of a cubic equation $a x^{3} + b x^{2} + c x + d = 0$ $ax^3+bx^2+cx+d=0$ is $b^{2} c^{2} - 4 a c^{3} - 4 b^{3} d - 27 a^{2} d^{2} + 18 a b c d$ $b^2c^2-4ac^3-4b^3d-27a^2d^2+18abcd$ .

If discriminant is positive, we have real roots;
if it is zero it has repeated roots and all real; and
if it is less than zero, we have one rel root and two complex conjugates.

In the given equation $7 x^{3} + x^{2} - 35 x - 5 = 0$ $7x^3+x^2-35x-5=0$ , discriminant is

$1^{2} \times {(- 35)}^{2} - 4 \times 7 \times {(- 35)}^{3} - 4 \times 1^{3} \times (- 5) - 27 \times 7^{2} \times {(- 5)}^{2} + 18 \times 7 \times 1 \times (- 35) \times (- 5)$ $1^2xx(-35)^2-4xx7xx(-35)^3-4xx1^3xx(-5)-27xx7^2xx(-5)^2+18xx7xx1xx(-35)xx(-5)$

= $1225 + 1200500 + 20 - 33075 + 22050 = 1190720$ $1225+1200500+20- 33075+22050=1190720$

Hence, we have real roots.

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Using the discriminant, give the nature of the roots of $7 x^{3} + x^{2} - 35 x = 5$ $7x^3+x^2-35x=5$ ?