Use completing square method to solve the equation, 3x2 - 7x + 2 = 0.

You're on a roll — Here are the solutions to the problems: 1. Use completing square method to solve the equation, 3x^2 - 7x + 2 = 0. Step 1: Divide the entire equation by the coefficient of x^2, which is 3. x^2 - (7)/(3)x + (2)/(3) = 0 Step 2: Move the constant term to the right side of the equation. x^2 - (7)/(3)x = -(2)/(3) Step 3: Complete the square on the left side. Take half of the coefficient of x, square it, and add it to both sides. The coefficient of x is -(7)/(3). Half of it is -(7)/(6). Squaring it gives (-(7)/(6))^2 = (49)/(36). x^2 - (7)/(3)x + (49)/(36) = -(2)/(3) + (49)/(36) Step 4: Factor the left side as a perfect square and simplify the right side. (x - (7)/(6))^2 = -(2 × 12)/(3 × 12) + (49)/(36) (x - (7)/(6))^2 = -(24)/(36) + (49)/(36) (x - (7)/(6))^2 = (25)/(36) Step 5: Take the square root of both sides. x - (7)/(6) = ±sqrt((25)/(36)) x - (7)/(6) = ±(5)/(6) Step 6: Solve for x. x = (7)/(6) ± (5)/(6) For the positive root: x_1 = (7)/(6) + (5)/(6) = (12)/(6) = 2 For the negative root: x_2 = (7)/(6) - (5)/(6) = (2)/(6) = (1)/(3) The solutions are x=2 or x=(1)/(3). 2. Solve for x, ((2)/(3))^2x × 36^x = 64. Step 1: Express all bases as powers of prime numbers or common bases. ((2)/(3))^2x × (6^2)^x = 2^6 ((2)/(3))^2x × 6^2x = 2^6 Step 2: Rewrite 6^2x as (2 × 3)^2x and apply the exponent to each factor. ((2)/(3))^2x × (2 × 3)^2x = 2^6 ((2)/(3))^2x × 2^2x × 3^2x = 2^6 Step 3: Distribute the exponent 2x in the first term. 2^2x3^2x × 2^2x × 3^2x = 2^6 Step 4: Cancel out the 3^2x terms. 2^2x × 2^2x = 2^6 Step 5: Use the exponent rule a^m × a^n = a^m+n. 2^2x+2x = 2^6 2^4x = 2^6 Step 6: Since the bases are equal, equate the exponents and solve for x. 4x = 6 x = (6)/(4) x = (3)/(2) The solution is x=(3)/(2). 3. The figure below shows a triangle PQR given that PQ = PR = 12cm, TR = 4cm, ST = 6cm and ST is parallel to QR. Calculate the perimeter of the quadrilateral QRTS. Step 1: Identify the given lengths and properties. PQ = 12 cm PR = 12 cm TR = 4 cm ST = 6 cm ST QR Step 2: Determine the length of PT. Since PR = PT + TR, we have: 12 cm = PT + 4 cm PT = 12 - 4 = 8 cm Step 3: Use the property of similar triangles. Since ST QR, triangle PST is similar to triangle PQR ( PST PQR). The ratio of corresponding sides is equal: (PS)/(PQ) = (PT)/(PR) = (ST)/(QR) Step 4: Calculate the ratio (PT)/(PR). (PT)/(PR) = 8 cm12 cm = (2)/(3) Step 5: Find the length of PS using the ratio. (PS)/(PQ) = (2)/(3) (PS)/(12 cm) = (2)/(3) PS = (2)/(3) × 12 cm = 8 cm Step 6: Find the length of SQ. Since PQ = PS + SQ, we have: 12 cm = 8 cm + SQ SQ = 12 - 8 = 4 cm Step 7: Find the length of QR using the ratio. (ST)/(QR) = (2)/(3) 6 cmQR = (2)/(3) 2 × QR = 6 cm × 3 2 QR = 18 cm QR = (18)/(2) = 9 cm Step 8: Calculate the perimeter of the quadrilateral QRTS. The perimeter is the sum of its sides: QR + RT + TS + SQ. Perimeter = 9 cm + 4 cm + 6 cm + 4 cm Perimeter = 23 cm The perimeter of quadrilateral QRTS is 23 cm. Send me the next one 📸