Solve the equation using the quadratic formula:
\[ x^2 - 2x - 24 = 0 \]
The quadratic formula is given by:
\[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \]
Identify the coefficients:
- \( a = 1 \)
- \( b = -2 \)
- \( c = -24 \)
Substitute these values into the formula and solve for \( x \).
Answer (3)
x 2 − 2 x − 24 = 0 a = 1 ; b = − 2 ; c = − 24 Δ = b 2 − 4 a c Δ = ( − 2 ) 2 − 4 ⋅ 1 ⋅ ( − 24 ) = 4 + 96 = 100 x 1 = 2 a − b − Δ ; x 2 = 2 a − b + Δ x 1 = 2 ⋅ 1 2 − 100 = 2 2 − 10 = 2 − 8 = − 4 x 2 = 2 ⋅ 1 2 + 100 = 2 2 + 10 = 2 12 = 6
There are another way to solve a quadratic equation we call, Sum and product.
Let's see how we can do this...
x 2 − 2 x − 24 = 0
S u m = − a b
P ro d u c t = a c
therefore
S u m = − 1 ( − 2 ) = 2
P ro d u c t = 1 − 24 = − 24
now we have to pick up 2 numbers that the sum should be 2 and the product should be -24, we just have to think a little.
Let's try -3 and 8, for example.
8 + ( − 3 ) = 5
8 ∗ ( − 3 ) = − 24
Doesn't work.
Let's try now 4 and -6.
4 + ( − 6 ) = − 2
4 ∗ ( − 6 ) = − 24
Can you see here, that we have to change the signal?!
therefore
Let's try -4 and 6
6 + ( − 4 ) = 2
6 ∗ ( − 4 ) = − 24
S u m = 2
and
P ro d u c t = − 24
Them it works.
x 1 = − 4 an d x 2 = 6
The quadratic equation x 2 − 2 x − 24 = 0 has solutions x = − 4 and x = 6 . This is determined by using the quadratic formula along with the coefficients identified from the equation. Calculating the discriminant showed that there are two real solutions to the equation.
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