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Solution - Solving quadratic inequalities using the quadratic formula

Interval notation - No Real Roots:

x \in (- \infty, \infty)

x∈(-∞,∞)

Solution:

x_{1} = i \cdot \sqrt{5}, x_{2} = - i \cdot \sqrt{5}

x_{1}=i\cdot\sqrt{5} , x_{2}=-i\cdot\sqrt{5}

See steps

Step-by-step explanation

1. Determine the quadratic inequality's coefficients $a$ , $b$ and $c$

The coefficients of our inequality, $x^{2} + 0 x + 5 > 0$ , are:

$a$ = 1

$b$ = 0

$c$ = 5

2. Plug these coefficients into the quadratic formula

To find the roots of a quadratic equation, plug its coefficients ( $a$ , $b$ and $c$ ) into the quadratic formula:

$x = (- b \pm s q r t (b^{2} - 4 a c)) / (2 a)$

$a = 1$
$b = 0$
$c = 5$

$x = (- 0 \pm s q r t (0^{2} - 4 * 1 * 5)) / (2 * 1)$

Simplify the exponents and square roots

$x = (- 0 \pm s q r t (0 - 4 * 1 * 5)) / (2 * 1)$

Perform any multiplication or division, from left to right:

$x = (- 0 \pm s q r t (0 - 4 * 5)) / (2 * 1)$

$x = (- 0 \pm s q r t (0 - 20)) / (2 * 1)$

Calculate any addition or subtraction, from left to right.

$x = (- 0 \pm s q r t (- 20)) / (2 * 1)$

Perform any multiplication or division, from left to right:

$x = (- 0 \pm s q r t (- 20)) / (2)$

to get the result:

$x = (- 0 \pm s q r t (- 20)) / 2$

3. Simplify square root $\sqrt{(- 20)}$

Simplify $- 20$ by finding its prime factors:

The prime factorization of $- 20$ is $2 i \cdot \sqrt{5}$

The square root of a negative number does not exist among the set of Real Numbers. We introduce The imaginary number "i", which is the square root of negative one. $\sqrt{(- 1)} = i$

$\sqrt{- 20} = \sqrt{(- 1) \cdot 20}$

$\sqrt{(- 1) \cdot 20} = i \sqrt{20}$

Write the prime factors:

$i \sqrt{20} = i \sqrt{2 \cdot 2 \cdot 5}$

Group the prime factors into pairs and rewrite them in exponent form:

$i \sqrt{2 \cdot 2 \cdot 5} = i \sqrt{2^{2} \cdot 5}$

Use the rule $\sqrt{(x^{2})} = x$ to simplify further:

$i \sqrt{2^{2} \cdot 5} = 2 i \cdot \sqrt{5}$

4. Solve the equation for x

$x = (- 0 \pm 2 i * s q r t (5)) / 2$

The ± means two roots are possible.

Separate the equations:
$x_{1} = (- 0 + 2 i * s q r t (5)) / 2$ and $x_{2} = (- 0 - 2 i * s q r t (5)) / 2$

$x_{1} = \frac{(0 + 2 i \cdot \sqrt{5})}{2}$

Simplify the arithmetic:

$x_{1} = \frac{2 i \cdot \sqrt{5}}{2}$

Simplify the fraction:

$x_{1} = i \cdot \sqrt{5}$

$x_{2} = \frac{(0 - 2 i \cdot \sqrt{5})}{2}$

Simplify the arithmetic:

$x_{2} = \frac{- 2 i \cdot \sqrt{5}}{2}$

Simplify the fraction:

$x_{2} = - i \cdot \sqrt{5}$

5. Find the intervals

Discriminant part of the quadratic formula:

$b^{2} - 4 a c < 0$ There are no real roots.
$b^{2} - 4 a c = 0$ There is one real root.
$b^{2} - 4 a c > 0$ There are two real roots.

Inequality function has no real roots, the parabola does not intersect with the x-axis. The quadratic formula requires taking the square root, and square root of negative number is not defined over the real line.

Interval is $(- \infty, \infty)$

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Why learn this

Whereas quadratic equations express the paths of arcs and the points along them, quadratic inequalities express the areas within and outside of these arcs and the ranges they cover. In other words, if quadratic equations tell us where the boundary is, then quadratic inequalities help us understand what we should focus on relative to that boundary. More practically, quadratic inequalities are used to create complex algorithms that fuel powerful software and to track how changes, such as prices at the grocery store, happen over time.

Terms and topics

Quadratic Inequalities

Solution - Solving quadratic inequalities using the quadratic formula

Step-by-step explanation

1. Determine the quadratic inequality's coefficients a, b and c