Question

1. (30-x)·x-(30-x+10)(x-6)=32 2. (2x-10)(x+8)-2x·x=220 3. 3(25-x)-2(25+x)=5 4. 2(10+x)+3·10=54 5. 2x= \frac{7}{4} (x+10)

Answer 1

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Answer 2

To solve the given equations, let's simplify each equation step by step and find the value of x.

Equation 1:

(30-x)·x - (30-x+10)(x-6) = 32

Expanding the equation: (30x - x^2) - (40 - x)(x - 6) = 32

Multiplying the terms inside the parentheses: 30x - x^2 - (40x - 6x - 6x + 36) = 32

Simplifying further: 30x - x^2 - (40x - 12x + 36) = 32 30x - x^2 - 40x + 12x - 36 = 32 - x^2 - 38x - 36 = 32 - x^2 - 38x - 36 - 32 = 0 - x^2 - 38x - 68 = 0

Rearranging the equation: x^2 + 38x + 68 = 0

Now, we can solve this quadratic equation using factoring, completing the square, or the quadratic formula.

Since the equation doesn't factor easily, let's use the quadratic formula to find the solutions for x:

The quadratic formula is given by: x = (-b ± sqrt(b^2 - 4ac))/(2a)

For our equation, a = 1, b = 38, and c = 68. Substituting these values into the quadratic formula:

x = (-38 ± sqrt(38^2 - 4 * 1 * 68))/(2 * 1)

Simplifying the equation: x = (-38 ± sqrt(1444 - 272))/2 x = (-38 ± sqrt(1172))/2 x = (-38 ± sqrt(4 * 293))/2 x = (-38 ± 2sqrt(293))/2

Simplifying further: x = -19 ± sqrt(293)

Therefore, the solutions for Equation 1 are: x = -19 + sqrt(293) and x = -19 - sqrt(293)

Equation 2:

(2x - 10)(x + 8) - 2x·x = 220

Expanding the equation: 2x^2 + 16x - 10x - 80 - 2x^2 = 220

Combining like terms: 6x - 80 = 220

Adding 80 to both sides: 6x = 300

Dividing both sides by 6: x = 50

Therefore, the solution for Equation 2 is: x = 50

Equation 3:

3(25 - x) - 2(25 + x) = 5

Simplifying the equation: 75 - 3x - 50 - 2x = 5

Combining like terms: -5x + 25 = 5

Subtracting 25 from both sides: -5x = -20

Dividing both sides by -5: x = 4

Therefore, the solution for Equation 3 is: x = 4

Equation 4:

2(10 + x) + 3 · 10 = 54

Simplifying the equation: 20 + 2x + 30 = 54

Combining like terms: 2x + 50 = 54

Subtracting 50 from both sides: 2x = 4

Dividing both sides by 2: x = 2

Therefore, the solution for Equation 4 is: x = 2

Equation 5:

2x + 7/4 + (x + 10) = 0

Combining like terms: 2x + x + 10 + 7/4 = 0

Adding the fractions: 2x + x + 10 + 7/4 = 0 2x + x + 10 + 1.75 = 0 3x + 11.75 = 0

Subtracting 11.75 from both sides: 3x = -11.75

Dividing both sides by 3: x = -11.75/3

Therefore, the solution for Equation 5 is: x = -11.75/3

To summarize, the solutions for the given equations are: Equation 1: x = -19 + sqrt(293) and x = -19 - sqrt(293) Equation 2: x = 50 Equation 3: x = 4 Equation 4: x = 2 Equation 5: x = -11.75/3