Solved on Mar 01, 2024

Solve the quadratic equation $2c^2 - 5c - 14 = -5$ for all real solutions in simplest form.

STEP 1

Assumptions
1. We are solving the quadratic equation for real solutions.
2. The equation is given in the form $2c^2 - 5c - 14 = -5$ .
3. The equation needs to be simplified before applying the quadratic formula or factoring.

STEP 2

First, we need to simplify the equation by moving all terms to one side to set the equation to zero.
$2c^2 - 5c - 14 + 5 = 0$

STEP 3

Combine like terms on the left side of the equation.
$2c^2 - 5c - 9 = 0$

STEP 4

Now, we need to determine if the quadratic equation can be factored. If not, we will use the quadratic formula to find the solutions.

STEP 5

Attempt to factor the quadratic equation. We are looking for two numbers that multiply to $2 \times (-9) = -18$ and add up to $-5$ .

STEP 6

After trying to find two such numbers, we realize that the quadratic equation cannot be factored easily. Therefore, we will use the quadratic formula to find the solutions.

STEP 7

Recall the quadratic formula, which is given by:
$c = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
where $a$ , $b$ , and $c$ are the coefficients from the quadratic equation $ax^2 + bx + c = 0$ .

STEP 8

Identify the coefficients from our equation $2c^2 - 5c - 9 = 0$ :
$a = 2, b = -5, c = -9$

STEP 9

Plug the coefficients into the quadratic formula.
$c = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(2)(-9)}}{2(2)}$

STEP 10

Simplify the terms inside the quadratic formula.
$c = \frac{5 \pm \sqrt{25 + 72}}{4}$

STEP 11

Combine the terms under the square root.
$c = \frac{5 \pm \sqrt{97}}{4}$

STEP 12

Since $\sqrt{97}$ cannot be simplified further, we have found the solutions in simplest radical form.
The solutions to the equation $2c^2 - 5c - 14 = -5$ are:
$c = \frac{5 + \sqrt{97}}{4} \quad \text{and} \quad c = \frac{5 - \sqrt{97}}{4}$

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Solve the quadratic equation $2c^2 - 5c - 14 = -5$ for all real solutions in simplest form.

STEP 1

Assumptions
1. We are solving the quadratic equation for real solutions.
2. The equation is given in the form $2c^2 - 5c - 14 = -5$ .
3. The equation needs to be simplified before applying the quadratic formula or factoring.

STEP 2

First, we need to simplify the equation by moving all terms to one side to set the equation to zero.
$2c^2 - 5c - 14 + 5 = 0$

STEP 3

Combine like terms on the left side of the equation.
$2c^2 - 5c - 9 = 0$

STEP 4

Now, we need to determine if the quadratic equation can be factored. If not, we will use the quadratic formula to find the solutions.

STEP 5

Attempt to factor the quadratic equation. We are looking for two numbers that multiply to $2 \times (-9) = -18$ and add up to $-5$ .

STEP 6

After trying to find two such numbers, we realize that the quadratic equation cannot be factored easily. Therefore, we will use the quadratic formula to find the solutions.

STEP 7

Recall the quadratic formula, which is given by:
$c = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
where $a$ , $b$ , and $c$ are the coefficients from the quadratic equation $ax^2 + bx + c = 0$ .

STEP 8

Identify the coefficients from our equation $2c^2 - 5c - 9 = 0$ :
$a = 2, b = -5, c = -9$

STEP 9

Plug the coefficients into the quadratic formula.
$c = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(2)(-9)}}{2(2)}$

STEP 10

Simplify the terms inside the quadratic formula.
$c = \frac{5 \pm \sqrt{25 + 72}}{4}$

STEP 11

Combine the terms under the square root.
$c = \frac{5 \pm \sqrt{97}}{4}$

STEP 12

Since $\sqrt{97}$ cannot be simplified further, we have found the solutions in simplest radical form.
The solutions to the equation $2c^2 - 5c - 14 = -5$ are:
$c = \frac{5 + \sqrt{97}}{4} \quad \text{and} \quad c = \frac{5 - \sqrt{97}}{4}$

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Solve the quadratic equation 2c2−5c−14=−52c^2 - 5c - 14 = -52c2−5c−14=−5 for all real solutions in simplest form.

STEP 1

Assumptions1. We are solving the quadratic equation for real solutions.2. The equation is given in the form 2c2−5c−14=−52c^2 - 5c - 14 = -52c2−5c−14=−5.3. The equation needs to be simplified before applying the quadratic formula or factoring.

STEP 2

First, we need to simplify the equation by moving all terms to one side to set the equation to zero.2c2−5c−14+5=02c^2 - 5c - 14 + 5 = 02c2−5c−14+5=0

STEP 3

Combine like terms on the left side of the equation.2c2−5c−9=02c^2 - 5c - 9 = 02c2−5c−9=0

STEP 4

Now, we need to determine if the quadratic equation can be factored. If not, we will use the quadratic formula to find the solutions.

STEP 5

Attempt to factor the quadratic equation. We are looking for two numbers that multiply to 2×(−9)=−182 \times (-9) = -182×(−9)=−18 and add up to −5-5−5.

STEP 6

After trying to find two such numbers, we realize that the quadratic equation cannot be factored easily. Therefore, we will use the quadratic formula to find the solutions.

STEP 7

Recall the quadratic formula, which is given by:c=−b±b2−4ac2ac = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}c=2a−b±b2−4ac​​where aaa, bbb, and ccc are the coefficients from the quadratic equation ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0.

STEP 8

Identify the coefficients from our equation 2c2−5c−9=02c^2 - 5c - 9 = 02c2−5c−9=0:a=2,b=−5,c=−9a = 2, b = -5, c = -9a=2,b=−5,c=−9

STEP 9

Plug the coefficients into the quadratic formula.c=−(−5)±(−5)2−4(2)(−9)2(2)c = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(2)(-9)}}{2(2)}c=2(2)−(−5)±(−5)2−4(2)(−9)​​

STEP 10

Simplify the terms inside the quadratic formula.c=5±25+724c = \frac{5 \pm \sqrt{25 + 72}}{4}c=45±25+72​​

STEP 11

Combine the terms under the square root.c=5±974c = \frac{5 \pm \sqrt{97}}{4}c=45±97​​

STEP 12

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Solve the quadratic equation $2c^2 - 5c - 14 = -5$ for all real solutions in simplest form.

Assumptions
1. We are solving the quadratic equation for real solutions.
2. The equation is given in the form $2c^2 - 5c - 14 = -5$ .
3. The equation needs to be simplified before applying the quadratic formula or factoring.

First, we need to simplify the equation by moving all terms to one side to set the equation to zero.
$2c^2 - 5c - 14 + 5 = 0$

Combine like terms on the left side of the equation.
$2c^2 - 5c - 9 = 0$

Attempt to factor the quadratic equation. We are looking for two numbers that multiply to $2 \times (-9) = -18$ and add up to $-5$ .

Recall the quadratic formula, which is given by:
$c = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
where $a$ , $b$ , and $c$ are the coefficients from the quadratic equation $ax^2 + bx + c = 0$ .

Identify the coefficients from our equation $2c^2 - 5c - 9 = 0$ :
$a = 2, b = -5, c = -9$

Plug the coefficients into the quadratic formula.
$c = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(2)(-9)}}{2(2)}$

Simplify the terms inside the quadratic formula.
$c = \frac{5 \pm \sqrt{25 + 72}}{4}$

Combine the terms under the square root.
$c = \frac{5 \pm \sqrt{97}}{4}$