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Math  /  Algebra

QuestionQuadratic Formula
Solve for x . x2−5x−24=0x=[?]\begin{array}{c} x^{2}-5 x-24=0 \\ x=[?] \end{array}
Remember the quadratic formula: x=−b±b2−4ac2ax=\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a}

Studdy Solution

STEP 1

What is this asking? We need to find the values of xx that make this specific quadratic equation equal to zero. Watch out! Don't forget there are *two* solutions to most quadratic equations, thanks to the plus-or-minus in the quadratic formula!

STEP 2

1. Identify Coefficients
2. Apply the Quadratic Formula
3. Simplify the Expression
4. Calculate the Solutions

STEP 3

Alright, let's **kick things off** by looking at our equation: x2−5x−24=0x^2 - 5x - 24 = 0.
Remember, the **standard form** of a quadratic equation is ax2+bx+c=0ax^2 + bx + c = 0.

STEP 4

So, matching things up, we can see that our **coefficients** are a=1a = 1, b=−5b = -5, and c=−24c = -24. **Crucially**, don't forget that negative sign in front of the 5 and 24!

STEP 5

Now, let's bring in the **star of the show**: the quadratic formula itself! x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

STEP 6

Let's **plug in** those coefficients we just found: a=1a = 1, b=−5b = -5, and c=−24c = -24. x=−(−5)±(−5)2−4⋅1⋅(−24)2⋅1x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4 \cdot 1 \cdot (-24)}}{2 \cdot 1}

STEP 7

Time to **clean things up**!
First, that double negative in front of the 5 becomes a positive 5. x=5±(−5)2−4⋅1⋅(−24)2⋅1x = \frac{5 \pm \sqrt{(-5)^2 - 4 \cdot 1 \cdot (-24)}}{2 \cdot 1}

STEP 8

Next, inside the square root, (−5)2(-5)^2 is (−5)⋅(−5)(-5) \cdot (-5), which gives us **positive** 25.
Then, 4⋅1⋅(−24)4 \cdot 1 \cdot (-24) gives us -96. x=5±25−(−96)2x = \frac{5 \pm \sqrt{25 - (-96)}}{2}

STEP 9

Subtracting a negative is the same as adding, so 25−(−96)25 - (-96) becomes 25+96=12125 + 96 = 121. x=5±1212x = \frac{5 \pm \sqrt{121}}{2}

STEP 10

The square root of 121 is 11, because 11⋅11=12111 \cdot 11 = 121. x=5±112x = \frac{5 \pm 11}{2}

STEP 11

Almost there!
Now we **split** the plus-or-minus into two separate calculations. x=5+112orx=5−112x = \frac{5 + 11}{2} \quad \text{or} \quad x = \frac{5 - 11}{2}

STEP 12

For the first one, 5+112=162=8\frac{5 + 11}{2} = \frac{16}{2} = 8. x=8x = 8

STEP 13

For the second one, 5−112=−62=−3\frac{5 - 11}{2} = \frac{-6}{2} = -3. x=−3x = -3

STEP 14

So, our **two solutions** are x=8x = 8 and x=−3x = -3!

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