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Solution - Solving quadratic equations using the quadratic formula

x_{1} = 2 + i

x_{1}=2+i

x_{2} = 2 - i

x_{2}=2-i

See steps

Step-by-step explanation

1. Find the coefficients

To find the coefficients, use the standard form of a quadratic equation:
$a x^{2} + b x + c = 0$
$x^{2} - 4 x + 5 = 0$

$a$ = 1

$b$ = -4

$c$ = 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:

7 additional steps

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

$a = 1$
$b = - 4$
$c = 5$

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

Simplify the exponents and square roots

$x = (- 1 * - 4 \pm s q r t (16 - 4 * 1 * 5)) / (2 * 1)$

Perform any multiplication or division, from left to right:

$x = (- 1 * - 4 \pm s q r t (16 - 4 * 5)) / (2 * 1)$

$x = (- 1 * - 4 \pm s q r t (16 - 20)) / (2 * 1)$

Calculate any addition or subtraction, from left to right.

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

Perform any multiplication or division, from left to right:

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

Perform any multiplication or division, from left to right:

$x = (4 \pm s q r t (- 4)) / 2$

to get the result:

$x = (4 \pm s q r t (- 4)) / 2$

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

Simplify $- 4$ by finding its prime factors:

The prime factorization of $- 4$ is $2 i$

3 additional steps

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

$\sqrt{- 4} = \sqrt{(- 1) \cdot 4}$

$\sqrt{(- 1) \cdot 4} = i \sqrt{4}$

Write the prime factors:

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

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

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

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

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

4. Solve the equation for x

$x = (4 \pm 2 i) / 2$

The ± means two answers are possible.

Separate the equations:
$x_{1} = (4 + 2 i) / 2$ and $x_{2} = (4 - 2 i) / 2$

3 additional steps

$x_{1} = \frac{(4 + 2 i)}{2}$

Break up the fraction:

$x_{1} = \frac{4}{2} + \frac{2 i}{2}$

Find the greatest common factor of the numerator and denominator:

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

Factor out and cancel the greatest common factor:

$x_{1} = 2 + \frac{2 i}{2}$

Simplify the fraction:

$x_{1} = 2 + i$

3 additional steps

$x_{2} = \frac{(4 - 2 i)}{2}$

Break up the fraction:

$x_{2} = \frac{4}{2} + \frac{- 2 i}{2}$

Find the greatest common factor of the numerator and denominator:

$x_{2} = \frac{(2 \cdot 2)}{(1 \cdot 2)} + \frac{- 2 i}{2}$

Factor out and cancel the greatest common factor:

$x_{2} = 2 + \frac{- 2 i}{2}$

Simplify the fraction:

$x_{2} = 2 - i$

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Why learn this

In their most basic function, quadratic equations define shapes like circles, ellipses and parabolas. These shapes can in turn be used to predict the curve of an object in motion, such as a ball kicked by football player or shot out of a cannon.

When it comes to an object’s movement through space, what better place to start than space itself—with the revolution of planets around the sun in our solar system. The quadratic equation was used to establish that planets’ orbits are elliptical, not circular. Determining the path and speed an object travels through space is possible even after it has come to a stop: the quadratic equation can calculate how fast a vehicle was moving when it crashed. With information like this, the automotive industry can design brakes to prevent collisions in the future. Many industries use the quadratic equation to predict and thus improve their products’ lifespan and safety.

Solution - Solving quadratic equations using the quadratic formula

Step-by-step explanation

1. Find the coefficients

2. Plug these coefficients into the quadratic formula

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

4. Solve the equation for x

Why learn this

Terms and topics

Related links

Solution - Solving quadratic equations using the quadratic formula

Step-by-step explanation

1. Find the coefficients

2. Plug these coefficients into the quadratic formula

3. Simplify square root (−4)

4. Solve the equation for x

Why learn this

Terms and topics

Related links

Latest Related Drills Solved

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