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Sagot :
To solve the given equation [tex]\( 10 - 9x^2 + 4x = -6x^2 \)[/tex], we will follow these steps:
1. Rewrite the equation in standard quadratic form:
Start by moving all terms to one side of the equation so that we have the form [tex]\( ax^2 + bx + c = 0 \)[/tex].
[tex]\[ 10 - 9x^2 + 4x + 6x^2 = 0 \][/tex]
Combine like terms:
[tex]\[ 10 - 3x^2 + 4x = 0 \][/tex]
This gives us the standard quadratic form:
[tex]\[ -3x^2 + 4x + 10 = 0 \][/tex]
2. Identify the coefficients [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
For the equation [tex]\(-3x^2 + 4x + 10 = 0\)[/tex], we have:
[tex]\[ a = -3, \quad b = 4, \quad c = 10 \][/tex]
3. Determining the discriminant:
The discriminant of a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] is given by [tex]\( \Delta = b^2 - 4ac \)[/tex].
Let's calculate the discriminant:
[tex]\[ \Delta = 4^2 - 4(-3)(10) \][/tex]
[tex]\[ \Delta = 16 + 120 \][/tex]
[tex]\[ \Delta = 136 \][/tex]
Since the discriminant is positive, we have two distinct real roots.
4. Finding the roots of the quadratic equation:
The roots of the quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] can be found using the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
Let's substitute the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( \Delta \)[/tex] into the quadratic formula:
[tex]\[ x = \frac{-4 \pm \sqrt{136}}{2(-3)} \][/tex]
[tex]\[ x = \frac{-4 \pm \sqrt{136}}{-6} \][/tex]
Simplify the expression under the square root:
[tex]\[ \sqrt{136} = \sqrt{4 \cdot 34} = 2\sqrt{34} \][/tex]
So the roots become:
[tex]\[ x = \frac{-4 \pm 2\sqrt{34}}{-6} \][/tex]
Simplify further by factoring out common terms in the numerator and denominator:
[tex]\[ x = \frac{2(-2 \pm \sqrt{34})}{-6} \][/tex]
[tex]\[ x = \frac{-2 \pm \sqrt{34}}{-3} \][/tex]
This gives us the solutions:
[tex]\[ x = \frac{2 - \sqrt{34}}{3} \quad \text{and} \quad x = \frac{2 + \sqrt{34}}{3} \][/tex]
Thus, the correct choice is (D):
[tex]\[ x = \frac{-2 \pm \sqrt{34}}{-3} \][/tex]
1. Rewrite the equation in standard quadratic form:
Start by moving all terms to one side of the equation so that we have the form [tex]\( ax^2 + bx + c = 0 \)[/tex].
[tex]\[ 10 - 9x^2 + 4x + 6x^2 = 0 \][/tex]
Combine like terms:
[tex]\[ 10 - 3x^2 + 4x = 0 \][/tex]
This gives us the standard quadratic form:
[tex]\[ -3x^2 + 4x + 10 = 0 \][/tex]
2. Identify the coefficients [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
For the equation [tex]\(-3x^2 + 4x + 10 = 0\)[/tex], we have:
[tex]\[ a = -3, \quad b = 4, \quad c = 10 \][/tex]
3. Determining the discriminant:
The discriminant of a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] is given by [tex]\( \Delta = b^2 - 4ac \)[/tex].
Let's calculate the discriminant:
[tex]\[ \Delta = 4^2 - 4(-3)(10) \][/tex]
[tex]\[ \Delta = 16 + 120 \][/tex]
[tex]\[ \Delta = 136 \][/tex]
Since the discriminant is positive, we have two distinct real roots.
4. Finding the roots of the quadratic equation:
The roots of the quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] can be found using the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
Let's substitute the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( \Delta \)[/tex] into the quadratic formula:
[tex]\[ x = \frac{-4 \pm \sqrt{136}}{2(-3)} \][/tex]
[tex]\[ x = \frac{-4 \pm \sqrt{136}}{-6} \][/tex]
Simplify the expression under the square root:
[tex]\[ \sqrt{136} = \sqrt{4 \cdot 34} = 2\sqrt{34} \][/tex]
So the roots become:
[tex]\[ x = \frac{-4 \pm 2\sqrt{34}}{-6} \][/tex]
Simplify further by factoring out common terms in the numerator and denominator:
[tex]\[ x = \frac{2(-2 \pm \sqrt{34})}{-6} \][/tex]
[tex]\[ x = \frac{-2 \pm \sqrt{34}}{-3} \][/tex]
This gives us the solutions:
[tex]\[ x = \frac{2 - \sqrt{34}}{3} \quad \text{and} \quad x = \frac{2 + \sqrt{34}}{3} \][/tex]
Thus, the correct choice is (D):
[tex]\[ x = \frac{-2 \pm \sqrt{34}}{-3} \][/tex]
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