how to find the values of x??
x^2 - 3x - 7 = 0
tried quadratic formula and completing the square what else can i do??
Either method should have given you the answer.
x = (1/2)[3 +/-sqrt37]
= 1.5 +/- 3.0414...
= 4.5414.. or -1.5414..
use the quadratic equation:
x= (3+-sqrt(9+28))/2=1.5+-1/2 sqrt 37
Now, completing the square:
x^2-3x=7
x^2-3x +9/4=7+9/4
(x-1.5)^2=37/4
x=1.5+-sqrt (37/4)
To find the values of x in the equation x^2 - 3x - 7 = 0, you have already mentioned two common methods - using the quadratic formula and completing the square. These methods should be sufficient to solve this quadratic equation. However, if you are looking for alternative approaches, you can also try factoring the quadratic equation.
Factoring involves breaking down the quadratic equation into two binomials that multiply together to give the original equation. Unfortunately, the quadratic x^2 - 3x - 7 = 0 cannot be factored easily with whole numbers.
Therefore, let's go ahead and solve the equation using the quadratic formula or completing the square.
Quadratic Formula Method:
The quadratic formula is a general formula for solving quadratic equations. For the equation ax^2 + bx + c = 0, the quadratic formula is:
x = (-b ± √(b^2 - 4ac)) / 2a
For the given equation x^2 - 3x - 7 = 0, a = 1, b = -3, and c = -7.
Substituting these values into the quadratic formula:
x = (-(-3) ± √((-3)^2 - 4(1)(-7))) / 2(1)
Simplifying further:
x = (3 ± √(9 + 28)) / 2
x = (3 ± √37) / 2
Therefore, the two solutions for x are:
x1 = (3 + √37) / 2
x2 = (3 - √37) / 2
Completing the Square Method:
Another way to solve this quadratic equation is by completing the square. The general steps for completing the square are:
1. Arrange the equation in the form: a(x^2 + bx) = -c.
2. Add (b/2)^2 to both sides of the equation to make it a perfect square.
3. Rewrite the left side as a perfect square and simplify the right side.
4. Take the square root of both sides and solve for x.
Let's apply these steps to the equation x^2 - 3x - 7 = 0:
1. Rearrange: x^2 - 3x = 7.
2. Add (3/2)^2 = 9/4 to both sides: x^2 - 3x + 9/4 = 7 + 9/4 = 37/4.
3. Rewrite left side as a perfect square: (x - 3/2)^2 = 37/4.
4. Take the square root: x - 3/2 = ±√(37/4).
Solve for x: x = 3/2 ± √(37/4).
Therefore, the two solutions for x are:
x1 = 3/2 + √(37/4)
x2 = 3/2 - √(37/4)
These are the methods available to find the values of x in the quadratic equation x^2 - 3x - 7 = 0.