The expression for ∠3 is x² + 2x, m∠3 is 48⁰ and ∠ SXP is 138⁰
How to find the value of x and ∠SXP
from the figure.
Given that QX is perpendicular to XS, then,
∠1 + ∠2 = 90⁰(complementary angles)
(x² + 2x) + (5x + 12) = 90⁰
x² + 7x + 12 = 9⁰
Subtract 90 from both sides to set the equation to zero:
x² + 7x + 12 - 90 = 0
Combine like terms: x² + 7x - 78 = 0
Factor the quadratic expression
: (x + 13)(x - 6) = 0
x + 13 = 0 orx - 6 = 0
x = -13 or x = 6
So, the solutions to the quadratic equation are (x = -13) and (x = 6).
Also,
PX is perpendicular to XR, then,
∠2 + ∠3 = 90⁰(complementary angles)
5x + 12 + ∠3 = 90⁰
Therefore
∠2 + ∠3 = ∠1 + ∠2
5x + 12 + ∠3 = (x² + 2x) + (5x + 12)
Solve for ∠3
∠3 = x² + 2x
Substitute x = 6 into the expressions
∠1 = x² + 2x
= 6² + 2(6)
= 36 + 12
= 48⁰
∠2 = 5x + 12
= 5(6) + 12
= 30 +12
= 42⁰
Then ∠3 is x² + 2x
∠SXP =∠1 + ∠2 + ∠3
= 48 + 42 +48
= 138⁰