Solve the equation: 9x^4 – 37x^2 + 4 = 0
The Correct Answer and Explanation is:
We are given the equation:9×4−37×2+4=09x^4 – 37x^2 + 4 = 09×4−37×2+4=0
To solve it, we can start by making a substitution to simplify the equation. Let:y=x2y = x^2y=x2
This substitution transforms the equation into a quadratic form in terms of yyy:9y2−37y+4=09y^2 – 37y + 4 = 09y2−37y+4=0
Now, we can solve this quadratic equation using the quadratic formula. The quadratic formula is given by:y=−b±b2−4ac2ay = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}y=2a−b±b2−4ac
For our equation, a=9a = 9a=9, b=−37b = -37b=−37, and c=4c = 4c=4. Substituting these values into the quadratic formula:y=−(−37)±(−37)2−4(9)(4)2(9)y = \frac{-(-37) \pm \sqrt{(-37)^2 – 4(9)(4)}}{2(9)}y=2(9)−(−37)±(−37)2−4(9)(4)
Simplifying the expression:y=37±1369−14418y = \frac{37 \pm \sqrt{1369 – 144}}{18}y=1837±1369−144y=37±122518y = \frac{37 \pm \sqrt{1225}}{18}y=1837±1225y=37±3518y = \frac{37 \pm 35}{18}y=1837±35
This gives two possible solutions for yyy:y=37+3518=7218=4y = \frac{37 + 35}{18} = \frac{72}{18} = 4y=1837+35=1872=4
ory=37−3518=218=19y = \frac{37 – 35}{18} = \frac{2}{18} = \frac{1}{9}y=1837−35=182=91
Now, recall that y=x2y = x^2y=x2, so we have two possible equations to solve for xxx:
- x2=4x^2 = 4×2=4
- x2=19x^2 = \frac{1}{9}x2=91
For the first equation, x2=4x^2 = 4×2=4, we take the square root of both sides:x=±2x = \pm 2x=±2
For the second equation, x2=19x^2 = \frac{1}{9}x2=91, we take the square root of both sides:x=±13x = \pm \frac{1}{3}x=±31
Thus, the solutions to the original equation are:x=2, x=−2, x=13, x=−13x = 2, \, x = -2, \, x = \frac{1}{3}, \, x = -\frac{1}{3}x=2,x=−2,x=31,x=−31
So, the solutions to the equation 9×4−37×2+4=09x^4 – 37x^2 + 4 = 09×4−37×2+4=0 are:x=2, x=−2, x=13, x=−13x = 2, \, x = -2, \, x = \frac{1}{3}, \, x = -\frac{1}{3}x=2,x=−2,x=31,x=−31
