i need to use the quadratic formula 2 solve these 4 questions:
1. x(squared)-6x+4=0
2. t(squared)+4t-2=0
3. 3x(squared)+10X+5=0
4. 2x=7-x(squared)
I just can't seem 2 get the right answer!
what is this
To solve these quadratic equations using the quadratic formula, you need to follow a few steps. The quadratic formula is:
x = (-b ± √(b^2 - 4ac)) / (2a)
Where a, b, and c are the coefficients of the quadratic equation in the standard form ax^2 + bx + c = 0.
Let's go through each question step by step:
1. x^2 - 6x + 4 = 0
In this equation: a = 1, b = -6, and c = 4.
Substitute these values into the quadratic formula:
x = (-(-6) ± √((-6)^2 - 4(1)(4))) / (2(1))
Simplifying the equation:
x = (6 ± √(36 - 16)) / 2
x = (6 ± √20) / 2
x = (6 ± 2√5) / 2
Simplify further by cancelling out common factors:
x = 3 ± √5
So, the solutions to this quadratic equation are x = 3 + √5 and x = 3 - √5.
Now, let's move on to the next equation:
2. t^2 + 4t - 2 = 0
Here, a = 1, b = 4, and c = -2.
Plug the values into the quadratic formula:
t = (-4 ± √(4^2 - 4(1)(-2))) / (2(1))
Simplifying:
t = (-4 ± √(16 + 8)) / 2
t = (-4 ± √24) / 2
t = (-4 ± 2√6) / 2
Simplify further:
t = -2 ± √6
So, the solutions to this equation are t = -2 + √6 and t = -2 - √6.
Moving on to the third equation:
3. 3x^2 + 10x + 5 = 0
Here, a = 3, b = 10, and c = 5.
Apply the quadratic formula:
x = (-10 ± √(10^2 - 4(3)(5))) / (2(3))
Simplifying:
x = (-10 ± √(100 - 60)) / 6
x = (-10 ± √40) / 6
x = (-10 ± 2√10) / 6
Simplify further:
x = -5/3 ± √10/3
Thus, the solutions to this equation are x = -5/3 + √10/3 and x = -5/3 - √10/3.
Finally, let's solve the fourth equation:
4. 2x = 7 - x^2
Rewrite the equation in standard form:
x^2 + 2x - 7 = 0
Here, a = 1, b = 2, and c = -7.
Plug the values into the quadratic formula:
x = (-2 ± √(2^2 - 4(1)(-7))) / (2(1))
Simplifying:
x = (-2 ± √(4 + 28)) / 2
x = (-2 ± √32) / 2
x = (-2 ± 4√2) / 2
Simplify further:
x = -1 ± 2√2
So, the solutions to this equation are x = -1 + 2√2 and x = -1 - 2√2.
Remember to always check the solutions by substituting them back into the original equation to verify if they satisfy it.
x^2 - 6 x + 4 = 0
x = [ 6 +/- sqrt (36 -16) ] / 2
x = 3 +/- 2 sqrt (9 - 4)
x = 3 +/-2 sqrt (5)
========================
t^2 + 4 t - 2 = 0
t = [ -4 +/- sqrt (16 + 8) ] / 2
t = [-4 +/- 2 sqrt (4+2) ] / 2
t = -2 +/- sqrt (6)
====================
3 x^2 + 10 x + 5 = 0
x = [ -10 +/- sqrt (100 - 60) ]/6
x = [ -10 +/- 2 sqrt (25 - 15) ]/6
x = -5/3 +/- (1/3) sqrt (10)