Question

what is the discriminant of:
$x {}^{2} + 4x - 2 = 0$
and
$- x { }^{2} + 10x - 25 = 0$
​

Answer 1

Quadratic Equations

A quadratic equation in a variable $x$ is an equation which is of the form $ax^2 + bx + c = 0$ where constants $a$, $b$ and $c$ are all real numbers and $a \neq 0$.

In case of a quadratic equation $ax^2 + bx + c = 0$ the expression $b^2 - 4ac$ is called the discriminant.

The given quadratic equations are;

1. ${x}^{2} + 4x - 2 = 0$

2. ${-x}^{2} + 10x - 25 = 0$

Solving for the first equation ${x}^{2} + 4x - 2 = 0$;

Comparing the given equation with the standard form of quadratic equation, we get:

$\qquad a = 1, \: b = 4, \: c = -2$

Now using the discriminant formula and solving the equation, we get:

$\implies (4)^2 - 4 \times 1 \times (-2) \\ \\ \implies 16 - 4 \times 2 \times (-2) \\ \\ \implies 16 - (-8) \\ \\ \implies 16 + 8 \\ \\ \implies \boxed{24}$

Hence, the required answer is 24.

Now solving for the second equation ${-x}^{2} + 10x - 25 = 0$;

Comparing the given equation with the standard form of quadratic equation, we get:

$\qquad a = -1, \: b = 10, \: c = -25$

Now using the discriminant formula and solving the equation, we get:

$\implies (10)^2 - 4 \times (-1) \times (-25) \\ \\ \implies 100 - 4 \times (-1) \times (-25) \\ \\ \implies 100 - (-4) \times (-25) \\ \\ \implies 100 - 100 \\ \\ \implies \boxed{0}$

Hence, the required answer is 0.

$\rule{90mm}{2pt}$

KNOWLEDGE BOOSTER

Let us consider a quadratic equation $ax^2 + bx + c = 0$, then nature of roots of quadratic equation depends upon Discriminant $D = b^2 - 4ac$ of the quadratic equation.

If $D = b^2 - 4ac > 0$, then roots of the equation are real and unequal.

If $D = b^2 - 4ac = 0$, then roots of the equation are real and equal.

If $D = b^2 - 4ac < 0$, then roots of the equation are unreal or complex or imaginary.

Answer 2

Answer:

Given :-

• x² + 4x - 2 = 0
• - x² + 10x - 25 = 0

To Find :-

• What is the discriminant of those equation.

Formula Used :-

$\bigstar$ Discriminant Formula :

$\mapsto \sf\boxed{\bold{\pink{Discriminant =\: b^2 - 4ac}}}$

Solution :-

$\sf\bold{\underline{\purple{\clubsuit\: In\: the\: case\: of\: x^2 + 4x - 2 =\: 0\: :-}}}$

Given Equation :

$\leadsto \bf x^2 + 4x - 2 =\: 0$

By comparing with ax² + bx + c = 0 we get,

◆ a = 1

◆ b = 4

◆ c = - 2

According to the question by using the formula we get,

$\implies \bf Discriminant =\: b^2 - 4ac$

$\implies \sf Discriminant =\: (4)^2 - 4(1)(- 2)$

$\implies \sf Discriminant =\: (4 \times 4) - 4 \times 1 \times (- 2)$

$\implies \sf Discriminant =\: 16 - 4 \times (- 2)$

$\implies \sf Discriminant =\: 16 - (- 8)$

$\implies \sf Discriminant =\: 16 + 8$

$\implies \sf\bold{\red{Discriminant =\: 24}}$

$\therefore$ The discriminant of x² + 4x - 2 = 0 is 24 .

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$\sf\bold{\underline{\purple{\clubsuit\: In\: the\: case\: of\: - x^2 + 10x - 25 =\: 0\: :-}}}$

Given Equation :

$\leadsto \bf - x^2 + 10x - 25 =\: 0$

By comparing with ax² + bx + c = 0 we get,

◆ a = - 1

◆ b = 10

◆ c = - 25

According to the question by using the formula we get,

$\mapsto \bf Discriminant =\: b^2 - 4ac$

$\mapsto \sf Discriminant =\: (10)^2 - 4(- 1)(- 25)$

$\mapsto \sf Discriminant =\: (10 \times 10) - 4 \times (- 1) \times (- 25)$

$\mapsto \sf Discriminant =\: 100 - (- 4) \times (- 25)$

$\mapsto \sf Discriminant =\: 100 + 4 \times (- 25)$

$\mapsto \sf Discriminant =\: 100 + (- 100)$

$\mapsto \sf Discriminant =\: 100 - 100$

$\mapsto \sf\bold{\red{Discriminant =\: 0}}$

$\therefore$ The discriminant of - x² + 10x - 25 = 0 is 0 .