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Solution - Solving quadratic inequalities using the quadratic formula

Solution:

1.536 < n < 8.464

1.536<n<8.464

Interval notation:

n \in (1.536, 8.464)

n∈(1.536,8.464)

See steps

Step-by-step explanation

1. Simplify the quadratic inequality into its standard form

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

Subtract $6$ from both sides of the inequality:

$n^{2} - 10 n + 19 < 6$

Subtract $6$ from both sides:

$n^{2} - 10 n + 19 - 6 < 6 - 6$

Simplify the expression

$n^{2} - 10 n + 13 < 0$

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

The coefficients of our inequality, $n^{2} - 10 n + 13 < 0$ , are:

$a$ = 1

$b$ = -10

$c$ = 13

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:

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

$a = 1$
$b = - 10$
$c = 13$

$n = (- 1 * - 10 \pm s q r t (- 10^{2} - 4 * 1 * 13)) / (2 * 1)$

Simplify the exponents and square roots

$n = (- 1 * - 10 \pm s q r t (100 - 4 * 1 * 13)) / (2 * 1)$

Perform any multiplication or division, from left to right:

$n = (- 1 * - 10 \pm s q r t (100 - 4 * 13)) / (2 * 1)$

$n = (- 1 * - 10 \pm s q r t (100 - 52)) / (2 * 1)$

Calculate any addition or subtraction, from left to right.

$n = (- 1 * - 10 \pm s q r t (48)) / (2 * 1)$

Perform any multiplication or division, from left to right:

$n = (- 1 * - 10 \pm s q r t (48)) / (2)$

Perform any multiplication or division, from left to right:

$n = (10 \pm s q r t (48)) / 2$

to get the result:

$n = (10 \pm s q r t (48)) / 2$

4. Simplify square root $\sqrt{(48)}$

Simplify $48$ by finding its prime factors:

Tree view of the prime factors of <math>48</math>:

The prime factorization of $48$ is $2^{4} \cdot 3$

Write the prime factors:

$\sqrt{48} = \sqrt{2 \cdot 2 \cdot 2 \cdot 2 \cdot 3}$

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

$\sqrt{2 \cdot 2 \cdot 2 \cdot 2 \cdot 3} = \sqrt{2^{2} \cdot 2^{2} \cdot 3}$

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

$\sqrt{2^{2} \cdot 2^{2} \cdot 3} = 2 \cdot 2 \cdot \sqrt{3}$

Perform any multiplication or division, from left to right:

$2 \cdot 2 \cdot \sqrt{3} = 4 \cdot \sqrt{3}$

5. Solve the equation for n

$n = (10 \pm 4 * s q r t (3)) / 2$

The ± means two roots are possible.

Separate the equations:
$n_{1} = (10 + 4 * s q r t (3)) / 2$ and $n_{2} = (10 - 4 * s q r t (3)) / 2$

$n_{1} = (10 + 4 * s q r t (3)) / 2$

Calculate the expression inside the parentheses

$n_{1} = (10 + 4 * s q r t (3)) / 2$

$n_{1} = (10 + 4 * 1.732) / 2$

Perform any multiplication or division, from left to right:

$n_{1} = (10 + 4 * 1.732) / 2$

$n_{1} = (10 + 6.928) / 2$

Calculate any addition or subtraction, from left to right.

$n_{1} = (10 + 6.928) / 2$

$n_{1} = (16.928) / 2$

Perform any multiplication or division, from left to right:

$n_{1} = \frac{16.928}{2}$

$n_{1} = 8.464$

$n_{2} = (10 - 4 * s q r t (3)) / 2$

Remove the parentheses

$n_{2} = (10 - 4 * s q r t (3)) / 2$

$n_{2} = (10 - 4 * 1.732) / 2$

Perform any multiplication or division, from left to right:

$n_{2} = (10 - 4 * 1.732) / 2$

$n_{2} = (10 - 6.928) / 2$

Calculate any addition or subtraction, from left to right.

$n_{2} = (10 - 6.928) / 2$

$n_{2} = (3.072) / 2$

Perform any multiplication or division, from left to right:

$n_{2} = \frac{3.072}{2}$

$n_{2} = 1.536$

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: 1.536, 8.464.

Since the $a$ coefficient is positive ( $a$ =1), this is a "positive" quadratic inequality and the parabola points upward, like a smile!

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 $n^{2} - 10 n + 13 < 0$ 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 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 (48)

5. Solve the equation for n

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 $\sqrt{(48)}$