how to solve algebra equation
√(5x^2 )+7x+2-√(4x^2+7x+18) =x-4
Ah, solving algebraic equations, the joy of every math lover. Now, let's dive into it.
First, let's simplify the equation to make it a bit more manageable.
We have √(5x^2) + 7x + 2 - √(4x^2 + 7x + 18) = x - 4.
Hmm, this equation is looking a little bit complicated. Let's try to make it less complicated by isolating the square roots.
To do that, we can move the terms with square roots to one side and all other terms to the other side.
So let's move the x terms to the left side of the equation and the constant terms to the right side. This will give us:
√(5x^2) - √(4x^2 + 7x + 18) - 7x + x = 4 - 2.
Now, we're almost there! All we need to do is simplify the equation further.
Using some algebraic sorcery, we can combine similar terms to get:
√(5x^2) - √(4x^2 + 7x + 18) - 6x = 2.
Now we're left with one square root term, which means we need to isolate it. Let's move all the other terms to the right side:
√(5x^2) - √(4x^2 + 7x + 18) = 6x + 2.
Almost done! To get rid of the square root term, we need to square both sides of the equation. This will happily eliminate the pesky square roots:
(√(5x^2) - √(4x^2 + 7x + 18))^2 = (6x + 2)^2.
Now, all that's left is some good old-fashioned multiplication and simplification. I'll leave you to it, my friend. Good luck solving the equation! And remember, algebra can be tricky, but you'll get through it. May the math gods be with you!
To solve the algebra equation √(5x^2 )+7x+2-√(4x^2+7x+18) =x-4, follow these steps:
Step 1: Simplify the square roots
- Rewrite the equation as √(5x^2 ) - √(4x^2+7x+18) + 7x + 2 = x - 4
- Simplify √(5x^2) to √(5) * √(x^2) = √(5) * x = √5x
- Simplify √(4x^2+7x+18) as much as possible
Step 2: Combine like terms
- Combine the terms that contain x: √5x - √(4x^2+7x+18) + 7x + 2 = x - 4
- Simplify the left side: (√5x + 7x) - √(4x^2+7x+18) + 2 = x - 4
- Combine like terms again: (√5x + 7x) - √(4x^2+7x+18) + 2 = x - 4
- Simplify: (7x + √5x) - √(4x^2+7x+18) + 2 = x - 4
Step 3: Isolate the radicals
- Move the terms with radicals to the other side of the equation: (7x + √5x) + 4 = x - 2 - √(4x^2+7x+18)
- Simplify the right side: (7x + √5x) + 4 = x - 2 - √(4x^2+7x+18)
Step 4: Square both sides of the equation
- Square both sides of the equation: [(7x + √5x) + 4]^2 = (x - 2 - √(4x^2+7x+18))^2
- Expand both sides: (7x + √5x)^2 + 2 * (7x + √5x) * 4 + 4^2 = (x - 2)^2 + 2 * (x - 2) * (-√(4x^2+7x+18)) + (-√(4x^2+7x+18))^2
- Simplify: (49x^2 + 14√5x^2 + 5x) + (56x + 8√5x) + 16 = x^2 - 4x + 4 + 2(x - 2)(-√(4x^2+7x+18)) + 4x^2 + 7x + 18
- Combine like terms: 49x^2 + 14√5x^2 + 5x + 56x + 8√5x + 16 = x^2 - 4x + 4 + 2(x - 2)(-√(4x^2+7x+18)) + 4x^2 + 7x + 18
Step 5: Simplify further
- Expand and simplify the right side: 49x^2 + 14√5x^2 + 5x + 56x + 8√5x + 16 = x^2 - 4x + 4 - 2(x^2 - 2x)(√(4x^2+7x+18)) + 4x^2 + 7x + 18
- Combine like terms again: 49x^2 + 14√5x^2 + 5x + 56x + 8√5x + 16 = x^2 - 4x + 4 - 2x^2√(4x^2+7x+18) + 4x^2 + 7x + 18
Step 6: Move all terms to one side of the equation
- Move all terms to one side of the equation: 49x^2 + 14√5x^2 + 5x + 56x + 8√5x + 16 - x^2 + 4x - 4 + 2x^2√(4x^2+7x+18) - 4x^2 - 7x - 18 = 0
- Simplify: 63x + 14√5x^2 + 10 + 2x^2√(4x^2+7x+18) - 4x^2 - 7x - 6 = 0
Step 7: Solve for x
- This equation is a quadratic equation in terms of x. Solve it using the quadratic formula or factoring techniques to find the values of x that satisfy the equation.
To solve the algebra equation √(5x^2) + 7x + 2 - √(4x^2 + 7x + 18) = x - 4, we can follow these steps:
Step 1: Simplify the radicals
First, simplify the radicals by finding the square roots.
√(5x^2) = √5 * √(x^2) = √5 * x = x√5
Similarly, simplify the other radical:
√(4x^2 + 7x + 18) = √(4x^2 + 4x + 3x + 18)
= √(4x(x + 1) + 3(x + 6))
= 2√(x(x + 1)) + √(3(x + 6))
Now, the equation becomes:
x√5 + 7x + 2 - (2√(x(x + 1)) + √(3(x + 6))) = x - 4
Step 2: Group the like terms
Group the terms with "x" and constants on one side of the equation:
x√5 + 7x - 2√(x(x + 1)) - √(3(x + 6)) = x - (2 + √(3(x + 6))) - 4
Step 3: Combine like terms
Combine the like terms on both sides of the equation:
x√5 + 7x - x = -6 - √(3(x + 6)) - 4 + 2√(x(x + 1))
6x√5 = -10 - √(3(x + 6)) + 2√(x(x + 1))
Step 4: Isolate the radical terms
To isolate the radical terms, move the other terms to the opposite side of the equation:
6x√5 + √(3(x + 6)) = -10 + 2√(x(x + 1))
Step 5: Square both sides
By squaring both sides of the equation, we can eliminate the radical terms:
(6x√5)^2 + 2(6x√5)√(3(x + 6)) + (√(3(x + 6)))^2 = (-10 + 2√(x(x + 1)))^2
Simplifying further:
36 * 5 * x^2 + 72 * x * √5 * √(3(x + 6)) + 3 * (x + 6) = 100 + 4 * (-10 * 2√(x(x + 1))) + (2√(x(x + 1)))^2
180x^2 + 72 * x * √15 * √(x + 6) + 3x + 18 = 100 - 80√(x(x + 1)) + 4(x(x + 1))
Step 6: Simplify and rearrange
Simplify both sides of the equation and rearrange terms to get the equation in standard form:
180x^2 + 72 * x * √15 * √(x + 6) + 3x + 18 = 100 - 80√(x(x + 1)) + 4x^2 + 4x
Combine the like terms:
176x^2 - 71x + 80√(x(x + 1)) + 72 * x * √15 * √(x + 6) = 82
Now, this is a quadratic equation in standard form. To solve it further, you can consider factoring, completing the square, or using the quadratic formula.
√(5x^2)+7x+2-√(4x^2+7x+18) = x-4
√(5x^2)-√(4x^2+7x+18) = -6x-2
now square both sides to get
(5x-2) - 2√(5x^2)(4x^2+7x+18) + (4x^2+7x+18) = (6x+2)^2
4x^2+12x+16 - 2√(20x^4+35x^3+90x^2) = 36x^2 + 24x + 4
√(20x^4+35x^3+90x^2) = -16x^2-6x+6
20x^4+35x^3+90x^2 = 256x^4+192x^3-156x^3-72x^2+36
236x^4+157x^3-246x^2-72x+36 = 0
quartics are tough, but solving graphically, I get
x = -1.23, -0.55, 0.29,0.81
Of these, discounting the spurious solutions introduced by squaring twice, I get only -0.55
better check my algebra