Solve the equation: x^4 – 10x^2 = -9
The Correct Answer and Explanation is:
To solve the equation x4−10×2=−9x^4 – 10x^2 = -9×4−10×2=−9, we can begin by making a substitution to simplify it.
- Substitution:
Let y=x2y = x^2y=x2. This way, the equation becomes: y2−10y=−9y^2 – 10y = -9y2−10y=−9 This is now a quadratic equation in terms of yyy. - Rearranging the equation:
We move all terms to one side to set the equation equal to zero: y2−10y+9=0y^2 – 10y + 9 = 0y2−10y+9=0 Now, we have a standard quadratic equation. - Solving the quadratic equation:
To solve the quadratic equation y2−10y+9=0y^2 – 10y + 9 = 0y2−10y+9=0, we can use the quadratic formula: y=−b±b2−4ac2ay = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}y=2a−b±b2−4ac For the equation y2−10y+9=0y^2 – 10y + 9 = 0y2−10y+9=0, the coefficients are a=1a = 1a=1, b=−10b = -10b=−10, and c=9c = 9c=9. Substituting these values into the quadratic formula: y=−(−10)±(−10)2−4(1)(9)2(1)y = \frac{-(-10) \pm \sqrt{(-10)^2 – 4(1)(9)}}{2(1)}y=2(1)−(−10)±(−10)2−4(1)(9) y=10±100−362y = \frac{10 \pm \sqrt{100 – 36}}{2}y=210±100−36 y=10±642y = \frac{10 \pm \sqrt{64}}{2}y=210±64 y=10±82y = \frac{10 \pm 8}{2}y=210±8 So, we have two possible solutions for yyy: y=10+82=9ory=10−82=1y = \frac{10 + 8}{2} = 9 \quad \text{or} \quad y = \frac{10 – 8}{2} = 1y=210+8=9ory=210−8=1 - Reversing the substitution:
Recall that y=x2y = x^2y=x2. So, we now substitute back:- For y=9y = 9y=9, we get x2=9x^2 = 9×2=9, which gives x=±3x = \pm 3x=±3.
- For y=1y = 1y=1, we get x2=1x^2 = 1×2=1, which gives x=±1x = \pm 1x=±1.
- Final solutions:
The solutions to the equation are x=3,−3,1,−1x = 3, -3, 1, -1x=3,−3,1,−1.
Thus, the solutions to the equation x4−10×2=−9x^4 – 10x^2 = -9×4−10×2=−9 are x=3,−3,1,−1x = 3, -3, 1, -1x=3,−3,1,−1.
