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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} = \frac{2}{3} + \frac{1}{3} i \cdot \sqrt{59}, x_{2} = \frac{2}{3} + \frac{- 1}{3} i \cdot \sqrt{59}

x_{1}=\frac{2}{3}+\frac{1}{3}i\cdot\sqrt{59} , x_{2}=\frac{2}{3}+\frac{-1}{3}i\cdot\sqrt{59}

See steps

Step-by-step explanation

1. Simplify the quadratic inequality into its standard form

$a x^{2} + b x + c \leq 0$

Subtract $23$ from both sides of the inequality:

$3 x^{2} - 4 x + 44 \leq 23$

Subtract $23$ from both sides:

$3 x^{2} - 4 x + 44 - 23 \leq 23 - 23$

Simplify the expression

$3 x^{2} - 4 x + 21 \leq 0$

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

The coefficients of our inequality, $3 x^{2} - 4 x + 21 \leq 0$ , are:

$a$ = 3

$b$ = -4

$c$ = 21

3. 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:

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

$a = 3$
$b = - 4$
$c = 21$

$x = (- 1 * - 4 \pm s q r t (- 4^{2} - 4 * 3 * 21)) / (2 * 3)$

Simplify the exponents and square roots

$x = (- 1 * - 4 \pm s q r t (16 - 4 * 3 * 21)) / (2 * 3)$

Perform any multiplication or division, from left to right:

$x = (- 1 * - 4 \pm s q r t (16 - 12 * 21)) / (2 * 3)$

$x = (- 1 * - 4 \pm s q r t (16 - 252)) / (2 * 3)$

Calculate any addition or subtraction, from left to right.

$x = (- 1 * - 4 \pm s q r t (- 236)) / (2 * 3)$

Perform any multiplication or division, from left to right:

$x = (- 1 * - 4 \pm s q r t (- 236)) / (6)$

Perform any multiplication or division, from left to right:

$x = (4 \pm s q r t (- 236)) / 6$

to get the result:

$x = (4 \pm s q r t (- 236)) / 6$

4. Simplify square root $\sqrt{(- 236)}$

Simplify $- 236$ by finding its prime factors:

The prime factorization of $- 236$ is $2 i \cdot \sqrt{59}$

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. $\sqrt{(- 1)} = i$

$\sqrt{- 236} = \sqrt{(- 1) \cdot 236}$

$\sqrt{(- 1) \cdot 236} = i \sqrt{236}$

Write the prime factors:

$i \sqrt{236} = i \sqrt{2 \cdot 2 \cdot 59}$

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

$i \sqrt{2 \cdot 2 \cdot 59} = i \sqrt{2^{2} \cdot 59}$

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

$i \sqrt{2^{2} \cdot 59} = 2 i \cdot \sqrt{59}$

5. Solve the equation for x

$x = (4 \pm 2 i * s q r t (59)) / 6$

The ± means two roots are possible.

Separate the equations:
$x_{1} = (4 + 2 i * s q r t (59)) / 6$ and $x_{2} = (4 - 2 i * s q r t (59)) / 6$

3 additional steps

$x_{1} = \frac{(4 + 2 i \cdot \sqrt{59})}{6}$

Break up the fraction:

$x_{1} = \frac{4}{6} + \frac{2 i \cdot \sqrt{59}}{6}$

Find the greatest common factor of the numerator and denominator:

$x_{1} = \frac{(2 \cdot 2)}{(3 \cdot 2)} + \frac{2 i \cdot \sqrt{59}}{6}$

Factor out and cancel the greatest common factor:

$x_{1} = \frac{2}{3} + \frac{2 i \cdot \sqrt{59}}{6}$

Simplify the fraction:

$x_{1} = \frac{2}{3} + \frac{1}{3} i \cdot \sqrt{59}$

3 additional steps

$x_{2} = \frac{(4 - 2 i \cdot \sqrt{59})}{6}$

Break up the fraction:

$x_{2} = \frac{4}{6} + \frac{- 2 i \cdot \sqrt{59}}{6}$

Find the greatest common factor of the numerator and denominator:

$x_{2} = \frac{(2 \cdot 2)}{(3 \cdot 2)} + \frac{- 2 i \cdot \sqrt{59}}{6}$

Factor out and cancel the greatest common factor:

$x_{2} = \frac{2}{3} + \frac{- 2 i \cdot \sqrt{59}}{6}$

Simplify the fraction:

$x_{2} = \frac{2}{3} + \frac{- 1}{3} i \cdot \sqrt{59}$

6. 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. Simplify the quadratic inequality into its standard form

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

3. Plug these coefficients into the quadratic formula

4. Simplify square root (−236)

5. Solve the equation for x

6. Find the intervals

Why learn this

Terms and topics

Related links

Latest Related Drills Solved

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

4. Simplify square root $\sqrt{(- 236)}$