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Solution - Solving quadratic inequalities using the quadratic formula

Solution:

x < - 1.165 or x > 1.165

x<-1.165 or x>1.165

Interval notation:

x \in (- \infty, - 1.165) ⋃ (1.165, \infty)

x∈(-∞,-1.165)⋃(1.165,∞)

See steps

Step-by-step explanation

1. Simplify the expression

30 additional steps

$(2 x^{2} - 4) \cdot (2 x^{2} - 4) < {(x^{2} - 1)}^{2}$

Expand the parentheses:

$2 x^{2} \cdot (2 x^{2} - 4) - 4 \cdot (2 x^{2} - 4) < {(x^{2} - 1)}^{2}$

Expand the parentheses:

$2 x^{2} \cdot 2 x^{2} + 2 x^{2} \cdot - 4 - 4 \cdot (2 x^{2} - 4) < {(x^{2} - 1)}^{2}$

Group like terms:

$(2 \cdot 2) \cdot (x^{2} \cdot x^{2}) + 2 x^{2} \cdot - 4 - 4 \cdot (2 x^{2} - 4) < {(x^{2} - 1)}^{2}$

Multiply the coefficients:

$4 \cdot (x^{2} \cdot x^{2}) + 2 x^{2} \cdot - 4 - 4 \cdot (2 x^{2} - 4) < {(x^{2} - 1)}^{2}$

Simplify the arithmetic:

$4 x^{4} + 2 x^{2} \cdot - 4 - 4 \cdot (2 x^{2} - 4) < {(x^{2} - 1)}^{2}$

Group like terms:

$4 x^{4} + (2 \cdot - 4) x^{2} - 4 \cdot (2 x^{2} - 4) < {(x^{2} - 1)}^{2}$

Multiply the coefficients:

$4 x^{4} - 8 x^{2} - 4 \cdot (2 x^{2} - 4) < {(x^{2} - 1)}^{2}$

Expand the parentheses:

$4 x^{4} - 8 x^{2} - 4 \cdot 2 x^{2} - 4 \cdot - 4 < {(x^{2} - 1)}^{2}$

Multiply the coefficients:

$4 x^{4} - 8 x^{2} - 8 x^{2} - 4 \cdot - 4 < {(x^{2} - 1)}^{2}$

Simplify the arithmetic:

$4 x^{4} - 8 x^{2} - 8 x^{2} + 16 < {(x^{2} - 1)}^{2}$

Combine like terms:

$4 x^{4} - 16 x^{2} + 16 < {(x^{2} - 1)}^{2}$

Expand the parentheses:

$4 x^{4} - 16 x^{2} + 16 < x^{2} \cdot (x^{2} - 1) - 1 \cdot (x^{2} - 1)$

Expand the parentheses:

$4 x^{4} - 16 x^{2} + 16 < x^{2} \cdot x^{2} + x^{2} \cdot - 1 - 1 \cdot (x^{2} - 1)$

Simplify the arithmetic:

$4 x^{4} - 16 x^{2} + 16 < x^{4} + x^{2} \cdot - 1 - 1 \cdot (x^{2} - 1)$

Expand the parentheses:

$4 x^{4} - 16 x^{2} + 16 < x^{4} - x^{2} - 1 x^{2} - 1 \cdot - 1$

Simplify the arithmetic:

$4 x^{4} - 16 x^{2} + 16 < x^{4} - x^{2} - 1 x^{2} + 1$

Group like terms:

$4 x^{4} - 16 x^{2} + 16 < x^{4} + (- x^{2} - x^{2}) + 1$

Simplify the arithmetic:

$4 x^{4} - 16 x^{2} + 16 < x^{4} - 2 x^{2} + 1$

Add $16$ to both sides:

$(4 x^{4} - 16 x^{2} + 16) + 2 x^{2} < (x^{4} - 2 x^{2} + 1) + 2 x^{2}$

Group like terms:

$4 x^{4} + (- 16 x^{2} + 2 x^{2}) + 16 < (x^{4} - 2 x^{2} + 1) + 2 x^{2}$

Simplify the arithmetic:

$4 x^{4} - 14 x^{2} + 16 < (x^{4} - 2 x^{2} + 1) + 2 x^{2}$

Group like terms:

$4 x^{4} - 14 x^{2} + 16 < x^{4} + (- 2 x^{2} + 2 x^{2}) + 1$

Simplify the arithmetic:

$4 x^{4} - 14 x^{2} + 16 < x^{4} + 1$

Subtract $16$ from both sides:

$(4 x^{4} - 14 x^{2} + 16) - x^{4} < (x^{4} + 1) - x^{4}$

Group like terms:

$(4 x^{4} - x^{4}) - 14 x^{2} + 16 < (x^{4} + 1) - x^{4}$

Simplify the arithmetic:

$3 x^{4} - 14 x^{2} + 16 < (x^{4} + 1) - x^{4}$

Group like terms:

$3 x^{4} - 14 x^{2} + 16 < (x^{4} - x^{4}) + 1$

Simplify the arithmetic:

$3 x^{4} - 14 x^{2} + 16 < 1$

Subtract $16$ from both sides:

$(3 x^{4} - 14 x^{2} + 16) - 16 < 1 - 16$

Simplify the arithmetic:

$3 x^{4} - 14 x^{2} < 1 - 16$

Simplify the arithmetic:

$3 x^{4} - 14 x^{2} < - 15$

Simplify the quadratic inequality into its standard form

$a x^{2} + b x + c < 0$

Add $- 15$ to both sides of the equation:

$- 14 x^{2} + 4 < - 15$

Add $- 15$ to both sides of the equation:

$- 14 x^{2} + 4 + 15 < - 15 + 15$

Simplify the expression

$- 14 x^{2} + 19 < 0$

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

The coefficients of our inequality, $- 14 x^{2} + 0 x + 19 < 0$ , are:

$a$ = -14

$b$ = 0

$c$ = 19

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:

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

$a = - 14$
$b = 0$
$c = 19$

$x = (- 0 \pm s q r t (0^{2} - 4 * - 14 * 19)) / (2 * - 14)$

Simplify the exponents and square roots

$x = (- 0 \pm s q r t (0 - 4 * - 14 * 19)) / (2 * - 14)$

Perform any multiplication or division, from left to right:

$x = (- 0 \pm s q r t (0 - - 56 * 19)) / (2 * - 14)$

$x = (- 0 \pm s q r t (0 - - 1064)) / (2 * - 14)$

Calculate any addition or subtraction, from left to right.

$x = (- 0 \pm s q r t (0 + 1064)) / (2 * - 14)$

$x = (- 0 \pm s q r t (1064)) / (2 * - 14)$

Perform any multiplication or division, from left to right:

$x = (- 0 \pm s q r t (1064)) / (- 28)$

to get the result:

$x = (- 0 \pm s q r t (1064)) / (- 28)$

4. Simplify square root $\sqrt{(1064)}$

Simplify $1064$ by finding its prime factors:

Tree view of the prime factors of <math>1064</math>:

The prime factorization of $1064$ is $2^{3} \cdot 7 \cdot 19$

Write the prime factors:

$\sqrt{1064} = \sqrt{2 \cdot 2 \cdot 2 \cdot 7 \cdot 19}$

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

$\sqrt{2 \cdot 2 \cdot 2 \cdot 7 \cdot 19} = \sqrt{2^{2} \cdot 2 \cdot 7 \cdot 19}$

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

$\sqrt{2^{2} \cdot 2 \cdot 7 \cdot 19} = 2 \cdot \sqrt{2 \cdot 7 \cdot 19}$

Perform any multiplication or division, from left to right:

$2 \cdot \sqrt{2 \cdot 7 \cdot 19} = 2 \cdot \sqrt{14 \cdot 19}$

$2 \cdot \sqrt{14 \cdot 19} = 2 \cdot \sqrt{266}$

5. Solve the equation for x

$x = (- 0 \pm 2 * s q r t (266)) / (- 28)$

The ± means two roots are possible.

Separate the equations:
$x_{1} = (- 0 + 2 * s q r t (266)) / (- 28)$ and $x_{2} = (- 0 - 2 * s q r t (266)) / (- 28)$

$x_{1} = (- 0 + 2 * s q r t (266)) / (- 28)$

Calculate the expression inside the parentheses

$x_{1} = (- 0 + 2 * s q r t (266)) / (- 28)$

$x_{1} = (- 0 + 2 * 16.31) / (- 28)$

Perform any multiplication or division, from left to right:

$x_{1} = (- 0 + 2 * 16.31) / (- 28)$

$x_{1} = (- 0 + 32.619) / (- 28)$

Calculate any addition or subtraction, from left to right.

$x_{1} = (- 0 + 32.619) / (- 28)$

$x_{1} = (32.619) / (- 28)$

Perform any multiplication or division, from left to right:

$x_{1} = \frac{32.619}{-} 28$

$x_{1} = - 1.165$

$x_{2} = (- 0 - 2 * s q r t (266)) / (- 28)$

$x_{2} = (- 0 - 2 * 16.31) / (- 28)$

Perform any multiplication or division, from left to right:

$x_{2} = (- 0 - 2 * 16.31) / (- 28)$

$x_{2} = (- 0 - 32.619) / (- 28)$

Calculate any addition or subtraction, from left to right.

$x_{2} = (- 0 - 32.619) / (- 28)$

$x_{2} = (- 32.619) / (- 28)$

Perform any multiplication or division, from left to right:

$x_{2} = - \frac{32.619}{-} 28$

$x_{2} = 1.165$

6. Find the intervals

To find the intervals of a quadratic inequality, we start by finding its parabola.

The roots of the parabola (where it meets the x-axis) are: -1.165, 1.165.

Since the $a$ coefficient is negative ( $a$ =-14), this is a "negative" quadratic inequality and the parabola points downward, like a frown.

If the inequality sign is ≤ or ≥ , then the intervals include the roots and we use a solid line. If the inequality sign is < or > the intervals do not include the roots and we use a dotted line.

7. Choose the correct interval (solution)

Since $- 14 x^{2} + 0 x + 19 < 0$ has a $<$ inequality sign, we look for the parabola intervals that are below the x-axis.

Solution:

Interval notation:

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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 expression

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

3. Plug these coefficients into the quadratic formula

4. Simplify square root (1064)

5. Solve the equation for x

6. Find the intervals

7. Choose the correct interval (solution)

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 $\sqrt{(1064)}$